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</div><h2>HL Paper 2</h2><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A continuous random variable \(X\) has a probability density function given by the </span><span style="font-family: times new roman,times; font-size: medium;">function \(f(x)\) , where</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {k{{\left( {x + 2} \right)}^2},}&{ - 2 \leqslant x < 0} \\ <br> {k,}&{0 \leqslant x \leqslant \frac{4}{3}} \\ <br> {0,}&{{\text{otherwise}}{\text{.}}} <br>\end{array}} \right.\]<br></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of \(k\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Hence find</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) the mean of \(X\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the median of \(X\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(k\int_{ - 2}^0 {{{\left( {x + 2} \right)}^2}} {\text{d}}x + \int_0^{\frac{4}{3}} {k{\text{d}}x} = 1\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{8k}}{3} + \frac{{4k}}{3} = 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = \frac{1}{4}\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Only ft on positive values of \(k\).</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \({\text{E}}(X) = \frac{1}{4}{\int_{ - 2}^0 {x\left( {x + 2} \right)} ^2}{\text{d}}x + \frac{1}{4}\int_0^{\frac{4}{3}} {x{\text{d}}x} \) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{1}{4} \times \frac{{ - 4}}{3} + \frac{2}{9}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = - \frac{1}{9}\) (\( - 0.111\)) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) median given by a such that \({\text{P}}(X < a) = 0.5\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{1}{4}{\int_{ - 2}^a {\left( {x + 2} \right)} ^2}{\text{d}}x = 0.5\) </span><strong><span style="font-family: times new roman,times; font-size: medium;">M1</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left[ {\frac{{{{\left( {x + 2} \right)}^3}}}{3}} \right]_{ - 2}^a = 2\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\left( {a + 2} \right)^3} - 0 = 6\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = \sqrt[3]{6} - 2\) (\(= - 0.183\)) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates recognised that integration was the appropriate technique to solve this question but the fact that the function was piecewise proved problematic for many. Good use of technology by some candidates was seen but few sketches of the function were made. A sketch would have been helpful to many candidates when attempting to solve (b (ii).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates recognised that integration was the appropriate technique to solve this question but the fact that the function was piecewise proved problematic for many. Good use of technology by some candidates was seen but few sketches of the function were made. A sketch would have been helpful to many candidates when attempting to solve (b (ii).</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">The events \(A\) and \(B\) </span>are such that \({\text{P}}(A) = 0.65\), \({\text{P}}(B) = 0.48\) and \({\text{P}}(A \cup B) = 0.818\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\text{P}}(A \cap B)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence show that the events \(A\) and \(B\) are independent.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>In Section A, where appropriate, accept answers that correctly round to 2 sf except in Q2, Q5(a) (ii), Q5(b) and Q8(a).</p>
<p class="p2"> </p>
<p class="p1">\(0.818 = 0.65 + 0.48 - {\text{P}}(A \cap B)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\({\text{P}}(A \cap B) = 0.312\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P}}(A)P(B) = 0.312\;\;\;( = 0.48 \times 0.65)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2">since <span class="s2">\({\text{P}}(A)P(B) = {\text{P}}(A \cap B)\) </span>then \(A\) and \(B\) are independent <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p3"> </p>
<p class="p2"><strong>Note: <span class="Apple-converted-space"> </span></strong>Only award the <strong><em>R1 </em></strong>if numerical values are seen. Award <strong><em>A1R1 </em></strong>for a correct conditional probability approach.</p>
<p class="p2"><em><strong>[2 marks]</strong></em></p>
<p class="p2"><em><strong>Total [4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The probability that the 08:00 train will be delayed on a work day (Monday to Friday) is \(\frac{1}{{10}}\). Assuming that delays occur independently,</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">find the probability that the 08:00 train is delayed exactly twice during any period of five work days;</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">find the minimum number of work days for which the probability of the 08:00 train being delayed at least once exceeds 90 %.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{B(5, 0.1)}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 2) = 0.0729\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \geqslant 1) = 1 - {\text{P}}(X = 0)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0.9 < 1 - {\left( {\frac{9}{{10}}} \right)^n}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n > \frac{{\ln 0.1}}{{\ln 0.9}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>n</em> = 22 days <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally answered successfully. Many candidates used the tabular feature of their GDC for (b) thereby avoiding potential errors in the algebraic manipulation of logs and inequalities.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally answered successfully. Many candidates used the tabular feature of their GDC for (b) thereby avoiding potential errors in the algebraic manipulation of logs and inequalities.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weight loss, in kilograms, of people using the slimming regime <em>SLIM3M</em> for a period of three months is modelled by a random variable <em>X</em>. Experimental data showed that 67 % of the individuals using <em>SLIM3M</em> lost up to five kilograms and 12.4 % lost at least seven kilograms. Assuming that X follows a normal distribution, find the expected weight loss of a person who follows the <em>SLIM3M</em> regime for three months.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{N}}(\mu ,{\text{ }}{\sigma ^2})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \leqslant 5) = 0.670 \Leftrightarrow \frac{{5 - \mu }}{\sigma } = 0.4399 \ldots \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X > 7) = 0.124 \Leftrightarrow \frac{{7 - \mu }}{\sigma } = 1.155 \ldots \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solve simultaneously</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu + 0.4399\sigma = 5{\text{ and }}\mu + 1.1552\sigma = 7\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu = 3.77{\text{ (3 sf)}}\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the expected weight loss is 3.77 kg</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A0</em></strong> for \(\mu = 3.78\) (answer obtained due to early rounding).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Although many candidates were successful in answering this question, a surprising number of candidates did not even attempt it. The main difficulty was in finding the correct z scores. A fairly common error was to misinterpret one of the conditions and obtain one of the equations as \(\frac{{7 - \mu }}{\sigma } = - 1.155 \ldots \). In some cases candidates failed to keep the accuracy throughout the question and obtained inaccurate answers.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X </em>has the distribution \({\text{B}}(30,{\text{ }}p)\) . Given that \({\text{E}}(X) = 10\) , find</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">the value of <em>p </em>;</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 10)\)</span> ;</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \geqslant 15)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(X) = np\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow 10 = 30p\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow p = \frac{1}{3}\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[1 mark]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(P(X = 10) = \left( {\begin{array}{*{20}{c}}<br> {30} \\ <br> {10} <br>\end{array}} \right){\left( {\frac{1}{3}} \right)^{10}}{\left( {\frac{2}{3}} \right)^{20}} = 0.153\) <strong> <em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \geqslant 15) = 1 - {\text{P}}(X \leqslant 14)\) <strong> <em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 1 - 0.9565... = 0.0435\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Again this proved to be an accessible question for students with many students gaining full marks. Most candidates used the calculator to find the answers to parts (b) and (c) which is what was intended, but candidates should be aware that there are often marks for recognising what needs to be found, even if the candidate does not obtain the final correct answer. It is suggested that in this style of question, candidates should indicate what they are trying to find as well as giving the final answer.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Again this proved to be an accessible question for students with many students gaining full marks. Most candidates used the calculator to find the answers to parts (b) and (c) which is what was intended, but candidates should be aware that there are often marks for recognising what needs to be found, even if the candidate does not obtain the final correct answer. It is suggested that in this style of question, candidates should indicate what they are trying to find as well as giving the final answer.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Again this proved to be an accessible question for students with many students gaining full marks. Most candidates used the calculator to find the answers to parts (b) and (c) which is what was intended, but candidates should be aware that there are often marks for recognising what needs to be found, even if the candidate does not obtain the final correct answer. It is suggested that in this style of question, candidates should indicate what they are trying to find as well as giving the final answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"> </p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The probability density function of a continuous random variable \(X\) is given by</p>
<p>\[f(x) = \left\{ {\begin{array}{*{20}{c}} {0,{\text{ }}x < 0} \\ {\frac{{\sin x}}{4},{\text{ }}0 \le x \le \pi } \\ {a(x - \pi ),{\text{ }}\pi < x \le 2\pi } \\ {0,{\text{ }}2\pi < x} \end{array}.} \right.\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Sketch the graph \(y = f(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\text{P}}(X \le \pi )\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(a = \frac{1}{{{\pi ^2}}}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the median of \(X\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Calculate the mean of \(X\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Calculate the variance of \(X\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\text{P}}\left( {\frac{\pi }{2} \le X \le \frac{{3\pi }}{2}} \right)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(\frac{\pi }{2} \le X \le \frac{{3\pi }}{2}\) find the probability that \(\pi \le X \le 2\pi \).</p>
<div class="marks">[4]</div>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><img src="images/Schermafbeelding_2016-01-06_om_08.40.33.png" alt></p>
<p class="p2">Award <strong><em>A1</em></strong> for sine curve from \(0\) to \(\pi \), award <strong><em>A1</em></strong> for straight line from \(\pi \) to \(2\pi \) <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\int_0^\pi {\frac{{\sin x}}{4}{\text{d}}x = \frac{1}{2}} \) <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>require \(\frac{1}{2} + \int_\pi ^{2\pi } {a(x - \pi ){\text{d}}x = 1} \) <strong><em>(M1)</em></strong></p>
<p>\( \Rightarrow \frac{1}{2} + a\left[ {\frac{{{{(x - \pi )}^2}}}{2}} \right]_\pi ^{2\pi } = 1\;\;\;\left( {{\text{or }}\frac{1}{2} + a\left[ {\frac{{{x^2}}}{2} - \pi x} \right]_\pi ^{2\pi } = 1} \right)\) <strong><em>A1</em></strong></p>
<p>\( \Rightarrow a\frac{{{\pi ^2}}}{2} = \frac{1}{2}\) <strong><em>A1</em></strong></p>
<p>\( \Rightarrow a = \frac{1}{{{\pi ^2}}}\) <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Must obtain the exact value. Do not accept answers obtained with calculator.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>\(0.5 + {\text{ area of triangle }} = 1\) <strong><em>R1</em></strong></p>
<p>area of triangle \( = \frac{1}{2}\pi \times a\pi = 0.5\) <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for correct use of area formula \( = 0.5\), <strong><em>A1</em></strong> for \(a\pi \).</p>
<p> </p>
<p>\(a = \frac{1}{{{\pi ^2}}}\) <strong><em>AG</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">median is \(\pi \) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\mu = \int_0^\pi {x \cdot \frac{{\sin x}}{4}{\text{d}}x + \int_\pi ^{2\pi } {x \cdot \frac{{x - \pi }}{{{\pi ^2}}}{\text{d}}x} } \) <strong><em>(M1)(A1)</em></strong></p>
<p>\( = 3.40339 \ldots = 3.40\;\;\;\left( {{\text{or }}\frac{\pi }{4} + \frac{{5\pi }}{6} = \frac{{13}}{{12}}\pi } \right)\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<p> </p>
<p> </p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>For \(\mu = 3.40339 \ldots \)</p>
<p><strong>EITHER</strong></p>
<p>\({\sigma ^2} = \int_0^\pi {{x^2} \cdot \frac{{\sin x}}{4}{\text{d}}x + \int_\pi ^{2\pi } {{x^2} \cdot \frac{{x - \pi }}{{{\pi ^2}}}{\text{d}}x - {\mu ^2}} } \) <strong><em>(M1)(A1)</em></strong></p>
<p><strong>OR</strong></p>
<p>\({\sigma ^2} = \int_0^\pi {{{(x - \mu )}^2} \cdot \frac{{\sin x}}{4}{\text{d}}x + \int_\pi ^{2\pi } {{{(x - \mu )}^2} \cdot \frac{{x - \pi }}{{{\pi ^2}}}{\text{d}}x} } \) <strong><em>(M1)(A1)</em></strong></p>
<p><strong>THEN</strong></p>
<p>\( = 3.866277 \ldots = 3.87\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\int_{\frac{\pi }{2}}^\pi {\frac{{\sin x}}{4}{\text{d}}x + \int_\pi ^{\frac{{3\pi }}{2}} {\frac{{x - \pi }}{{{\pi ^2}}}{\text{d}}x = 0.375\;\;\;\left( {{\text{or }}\frac{1}{4} + \frac{1}{8} = \frac{3}{8}} \right)} } \) <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}\left( {\pi \le X \le 2\pi \left| {\frac{\pi }{2} \le X \le \frac{{3\pi }}{2}} \right.} \right) = \frac{{{\text{P}}\left( {\pi \le X \le \frac{{3\pi }}{2}} \right)}}{{{\text{P}}\left( {\frac{\pi }{2} \le X \le \frac{{3\pi }}{2}} \right)}}\) <strong><em>(M1)(A1)</em></strong></p>
<p>\( = \frac{{\int_\pi ^{\frac{{3\pi }}{2}} {\frac{{(x - \pi )}}{{{\pi ^2}}}{\text{d}}x} }}{{0.375}} = \frac{{0.125}}{{0.375}}\;\;\;\left( {{\text{or }} = \frac{{\frac{1}{8}}}{{\frac{3}{8}}}{\text{ from diagram areas}}} \right)\) <strong><em>(M1)</em></strong></p>
<p>\( = \frac{1}{3}\;\;\;(0.333)\) <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<p><strong><em>Total [20 marks]</em></strong></p>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates sketched the graph correctly. In a few cases candidates did not seem familiar with the shape of the graphs and ignored the fact that the graph represented a pdf. The correct sketch assisted greatly in the rest of the question.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates answered this question correctly.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">A few good proofs were seen but also many poor answers where the candidates assumed what you were trying to prove and verified numerically the result.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates stated the value correctly but many others showed no understanding of the concept.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many candidates scored full marks in this question; many others could not apply the formula due to difficulties in dealing with the piecewise function. For example, a number of candidates divided the final answer by two.</p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many misconceptions were identified: use of incorrect formula (e.g. formula for discrete distributions), use of both expressions as integrand and division of the result by 2 at the end.</p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This part was fairly well done with many candidates achieving full marks.</p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many candidates had difficulties with this part showing that the concept of conditional probability was poorly understood. The best candidates did it correctly from the sketch.</p>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A ski resort finds that the mean number of accidents on any given weekday (Monday to Friday) is 2.2 . The number of accidents can be modelled by a Poisson distribution.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that in a certain week (Monday to Friday only)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) there are fewer than 12 accidents;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) there are more than 8 accidents, given that there are fewer than 12 accidents.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a(i)(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Due to the increased usage, it is found that the probability of more than 3 accidents in a day at the weekend (Saturday and Sunday) is 0.24.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Assuming a Poisson model,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) calculate the mean number of accidents per day at the weekend (Saturday and Sunday);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) calculate the probability that, in the four weekends in February, there will be more than 5 accidents during at least two of the weekends.</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b(i)(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">It is found that 20 % of skiers having accidents are at least 25 years of age and 40 % </span></span></span><span style="font-family: 'times new roman', times; font-size: medium;">are under 18 years of age.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Assuming that the ages of skiers having accidents are normally distributed, find the mean age of skiers having accidents.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(X \sim {\text{Po}}(11)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \leqslant 11) = 0.579\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\text{P}}(X > 8\left| {x < 12) = } \right.\) <strong><em>(M1)</em></strong> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{{\text{P}}(8 < X < 12)}}{{{\text{P}}(X < 12)}}{\text{ }}\left( {{\text{or }}\frac{{{\text{P}}(X \leqslant 11) - {\text{P}}(X \leqslant 8)}}{{{\text{P}}(X \leqslant 11)}}{\text{ or }}\frac{{0.3472...}}{{0.5792...}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.600\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[6 marks]</span><br></em></strong></p>
<div class="question_part_label">a(i)(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(Y \sim {\text{Po}}(m)\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Y > 3) = 0.24\) <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Y \leqslant 3) = 0.76\) <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{e}}^{ - m}}\left( {1 + m + \frac{1}{2}{m^2} + \frac{1}{6}{m^3}} \right) = 0.76\) <strong><em>(A1)</em></strong></span> </p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>At most two of the above lines can be implied.</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">Attempt to solve equation with GDC </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)</em></strong></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(m = 2.49\) <strong><em>A1</em></strong></span></p>
<p> </p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(A \sim {\text{Po}}(4.98)\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(A > 5) = 1 - {\text{P}}(A \leqslant 5) = 0.380...\) <strong><em>M1A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(W \sim {\text{B}}(4,\,0.380...)\) <strong><em>(M1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(W \geqslant 2) = 1 - {\text{P}}(W \leqslant 1) = 0.490\) <strong><em>M1A1</em></strong></span></p>
<p> <strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;"><em>[10 marks]</em></strong></p>
<p> </p>
<div class="question_part_label">b(i)(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(A < 25) = 0.8,{\text{ P}}(A < 18) = 0.4\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{25 - \mu }}{\sigma } = 0.8416...\) <strong><em>(M1)(A1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{18 - \mu }}{\sigma } = -0.2533...{\text{ (or}} -0.2534{\text{ from tables)}}\) <strong><em>(M1)(A1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">solving these equations <strong><em>(M1)</em></strong></span> </p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu = 19.6\) <strong><em>A1</em></strong></span> </p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Accept just 19.6, 19 or 20; award A0 to any other final answer.</span></p>
<p><strong style="font-size: 11px; line-height: normal;"><em><span style="font-family: 'times new roman', times; font-size: medium;"> </span></em></strong></p>
<p><strong style="font-size: 11px; line-height: normal;"><em><span style="font-family: 'times new roman', times; font-size: medium;">[6 marks]</span></em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Generally, candidates had difficulties with this question, mainly in applying conditional probability and interpreting the expressions ‘more than’, ‘at least’ and ‘under’ to obtain correct expressions. Although many candidates identified the binomial distribution in part (b) (ii), very few succeeded in answering this question due to incorrect interpretation of the question or due to accuracy errors.</span></p>
<div class="question_part_label">a(i)(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: small;">Generally, candidates had difficulties with this question, mainly in applying conditional probability and interpreting the expressions ‘more than’, ‘at least’ and ‘under’ to obtain correct expressions. Although many candidates identified the binomial distribution in part (b) (ii), very few succeeded in answering this question due to incorrect interpretation of the question or due to accuracy errors.</span></p>
<div class="question_part_label">b(i)(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Generally, candidates had difficulties with this question, mainly in applying conditional probability and interpreting the expressions ‘more than’, ‘at least’ and ‘under’ to obtain correct expressions. Although many candidates identified the binomial distribution in part (b) (ii), very few succeeded in answering this question due to incorrect interpretation of the question or due to accuracy errors.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The probability density function of a continuous random variable <em>X </em>is given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(f(x) = \frac{1}{{1 + {x^4}}}\), \(0\) \(''\) \(x\) \(''\) \(a\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of <em>a </em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the mean of <em>X </em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_0^a {\frac{1}{{1 + {x^4}}}{\text{d}}x = 1} \) <strong> <em>M2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>a</em> = 1.40 <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(X) = \int_0^a {\frac{x}{{1 + {x^4}}}{\text{d}}x} \) <strong> <em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( { = \frac{1}{2}\arctan ({a^2})} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.548 <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates picked up some marks for this question, but only a few gained full marks. In part (a) many candidates did not appreciate the need for the calculator to find a value of <em>a</em>. Candidates had more success with part (b) with a number of candidates picking up follow through marks.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates picked up some marks for this question, but only a few gained full marks. In part (a) many candidates did not appreciate the need for the calculator to find a value of <em>a</em>. Candidates had more success with part (b) with a number of candidates picking up follow through marks.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The length, <em>X</em> metres, of a species of fish has the probability density function</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{r}}<br> {a{x^2},}&{{\text{for }}0 \leqslant x \leqslant 0.5} \\ <br> {0.5a(1 - x),}&{{\text{for }}0.5 \leqslant x \leqslant 1} \\ <br> {0,}&{{\text{otherwise }}{\text{.}}} <br>\end{array}} \right.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>a</em> = 9.6.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the graph of the distribution.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 35.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({\text{P}}(X < 0.6)\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_0^{0.5} {a{x^2}{\text{d}}x + \int_{0.5}^1 {0.5a(1 - x){\text{d}}x = 1} } \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{5a}}{{48}}\) (or equivalent) or \(a \times 0.104 \ldots = 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for considering two definite integrals.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Award <strong><em>A1</em></strong> for equating to 1.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Award <strong><em>A1</em></strong> for a correct equation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The <strong><em>A1A1</em></strong> can be awarded in any order.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><em style="font-family: 'times new roman', times; font-size: medium;"> </em></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><em style="font-family: 'times new roman', times; font-size: medium;">a</em><span style="font-family: 'times new roman', times; font-size: medium;"> = 9.6 </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>AG</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">correct shape for \(0 \leqslant x \leqslant 0.5\) and \(f(0.5) \approx 2.4\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">correct shape for \(0.5 \leqslant x \leqslant 1\) and \(f(1) = 0\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to find \({\text{P}}(X < 0.6)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">direct GDC use or <em>eg</em> \({\text{P}}(0 \leqslant X \leqslant 0.5) + {\text{P}}(0.5 \leqslant X \leqslant 0.6)\) or \(1 - {\text{P}}(0.6 \leqslant X \leqslant 1)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X < 0.6) = 0.616{\text{ }}\left( { = \frac{{77}}{{125}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was generally well done. Common errors usually involved not recognizing that the sum of the two integrals was equal to one, premature rounding or not showing full working to conclusively show that <em>a </em>= 9.6.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (b) was not well done with many graphs poorly labelled and offering no reference to domain and range.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (c) was reasonably well done. The most common error involved calculating an incorrect probability from an incorrect definite integral.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The number of accidents that occur at a large factory can be modelled by a Poisson distribution with a mean of 0.5 accidents per month.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that no accidents occur in a given month.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that no accidents occur in a given 6 month period.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the length of time, in complete months, for which the probability that at least 1 accident occurs is greater than 0.99.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">To encourage safety the factory pays a bonus of $1000 into a fund for workers if no accidents occur in any given month, a bonus of $500 if 1 or 2 accidents occur and no bonus if more than 2 accidents occur in the month.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Calculate the expected amount that the company will pay in bonuses each month.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the probability that in a given 3 month period the company pays a total of exactly $2000 in bonuses.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">P(<em>x</em> = 0) = 0.607 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using \(X \sim {\text{Po}}(3)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using \({(0.6065…)^6}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">P(<em>X</em> = 0) = 0.0498 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{Po}}(0.5t)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(x \geqslant 1) = 1 - {\text{P}}(x = 0)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(x = 0) < 0.01\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{e}}^{ - 0.5t}} < 0.01\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( - 0.5t < \ln (0.01)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(t > 9.21{\text{ months}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore 10 months <strong><em>A1N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Full marks can be awarded for answers obtained directly from GDC if a systematic method is used and clearly shown.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) P(1 or 2 accidents) = 0.37908… <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(B) = 1000 \times 0.60653... + 500 \times 0.37908…\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \$ 796\,\,\,\,\,\)(accept $797 or $796.07) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) P(2000) = P(1000, 1000, 0) + P(1000, 0, 1000) + P(0, 1000, 1000) +</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">P(1000, 500, 500) + P(500, 1000, 500) + P(500, 500, 1000) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for noting that 2000 can be written both as \(2 \times 1000 + 1 \times 0\) and \(2 \times 500 + 1 \times 1000\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 3{(0.6065...)^2}(0.01437...) + 3{(0.3790...)^2}(0.6065…)\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.277\,\,\,\,\,\)(accept 0.278) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates successfully answered (a) and (b). Although many found the correct answer to (c), communication of their reasoning was weak. This was also true for (d)(i). Answers to (d)(ii) were mostly scrappy and rarely worthy of credit.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates successfully answered (a) and (b). Although many found the correct answer to (c), communication of their reasoning was weak. This was also true for (d)(i). Answers to (d)(ii) were mostly scrappy and rarely worthy of credit.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates successfully answered (a) and (b). Although many found the correct answer to (c), communication of their reasoning was weak. This was also true for (d)(i). Answers to (d)(ii) were mostly scrappy and rarely worthy of credit.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates successfully answered (a) and (b). Although many found the correct answer to (c), communication of their reasoning was weak. This was also true for (d)(i). Answers to (d)(ii) were mostly scrappy and rarely worthy of credit.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The number of visitors that arrive at a museum every minute can be modelled by a Poisson distribution with mean 2.2.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If the museum is open 6 hours daily, find the expected number of visitors in 1 day.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that the number of visitors arriving during an hour exceeds 100.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that the number of visitors in each of the 6 hours the museum is open exceeds 100.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The ages of the visitors to the museum can be modelled by a normal distribution </span><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;">with mean \(\mu \) and variance \({\sigma ^2}\) . The records show that 29 % of the visitors are under </span><span style="font-family: 'times new roman', times; font-size: medium;">35 years of age and 23 % are at least 55 years of age.</span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the values of \(\mu \) and \(\sigma \) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">The ages of the visitors to the museum can be modelled by a normal distribution with mean \(\mu \) and variance \({\sigma ^2}\) . The records show that 29 % of the visitors are under 35 years of age and 23 % are at least 55 years of age.</span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">One day, 100 visitors under 35 years of age come to the museum. Estimate the number of visitors under 50 years of age that were at the museum on that day.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2.2 \times 6 \times 60 = 792\) <em><strong>(M1)A1</strong></em><br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> <span style="font-family: 'times new roman', times; font-size: medium;">\(V \sim {\text{Po}}(2.2 \times 60)\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(V > 100) = 0.998\) <em><strong>(M1)A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({(0.997801...)^6} = 0.987\) <em><strong>(M1)A1</strong></em></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"><em><strong>[2 marks]</strong></em></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"> </p>
<p style="font-family: 'times new roman', times; font-size: medium; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(A \sim {\text{N}}(\mu ,{\text{ }}{\sigma ^2})\)</span></p>
<p style="font-family: 'times new roman', times; font-size: medium; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(A < 35) = 0.29{\text{ and P}}(A > 55) = 0.23 \Rightarrow {\text{P}}(A < 55) = 0.77\)</span></p>
<p style="font-family: 'times new roman', times; font-size: medium; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}\left( {Z < \frac{{35 - \mu }}{\sigma }} \right) = 0.29{\text{ and P}}\left( {Z < \frac{{55 - \mu }}{\sigma }} \right) = 0.77\) <em style="font-style: italic;"><strong style="font-weight: bold;">(M1)</strong></em></span><span style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font-size: 14px; line-height: 20px;"> </span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">use of inverse normal <strong><em>(M1)</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{35 - \mu }}{\sigma } = - 0.55338...{\text{ and }}\frac{{55 - \mu }}{\sigma } = 0.738846...\) <em style="font-style: italic;"> <strong>(A1)</strong></em></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">solving simultaneously <strong> <em>(M1)</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu = 43.564...{\text{ and }}\sigma = 15.477...\) <strong> <em>A1A1</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu = 43.6{\text{ and }}\sigma = 15.5{\text{ (3sf)}}\)</span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0.29n = 100 \Rightarrow n = 344.82...\) <em style="font-style: italic;"><strong style="font-weight: bold;">(M1)(A1)</strong></em><br></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(A < 50) = 0.66121...\) <em style="font-style: italic;"><strong style="font-weight: bold;">(A1)</strong></em></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">expected number of visitors under <span style="font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">50 = 228</span> </span><strong><em>(M1)A1</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally well done by most candidates. It was evident that candidates had been well prepared in Poisson and normal distribution. In parts (a)-(d) candidates were usually successful and appropriate methods were shown although many candidates used labored algebraic approaches to solving simultaneous equations and wasted time answering part (d). Part (e) was very well answered by a smaller number of candidates but it was obviously more demanding in its level of abstraction.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally well done by most candidates. It was evident that candidates had been well prepared in Poisson and normal distribution. In parts (a)-(d) candidates were usually successful and appropriate methods were shown although many candidates used labored algebraic approaches to solving simultaneous equations and wasted time answering part (d). Part (e) was very well answered by a smaller number of candidates but it was obviously more demanding in its level of abstraction.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally well done by most candidates. It was evident that candidates had been well prepared in Poisson and normal distribution. In parts (a)-(d) candidates were usually successful and appropriate methods were shown although many candidates used labored algebraic approaches to solving simultaneous equations and wasted time answering part (d). Part (e) was very well answered by a smaller number of candidates but it was obviously more demanding in its level of abstraction.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally well done by most candidates. It was evident that candidates had been well prepared in Poisson and normal distribution. In parts (a)-(d) candidates were usually successful and appropriate methods were shown although many candidates used labored algebraic approaches to solving simultaneous equations and wasted time answering part (d). Part (e) was very well answered by a smaller number of candidates but it was obviously more demanding in its level of abstraction.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally well done by most candidates. It was evident that candidates had been well prepared in Poisson and normal distribution. In parts (a)-(d) candidates were usually successful and appropriate methods were shown although many candidates used labored algebraic approaches to solving simultaneous equations and wasted time answering part (d). Part (e) was very well answered by a smaller number of candidates but it was obviously more demanding in its level of abstraction.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>A continuous random variable \(X\) has probability density function \(f\) given by</p>
<p>\(f(x) = \left\{ {\begin{array}{*{20}{l}} {\frac{{{x^2}}}{a} + b,}&{0 \leqslant x \leqslant 4} \\ 0&{{\text{otherwise}}} \end{array}} \right.{\text{where }}a{\text{ and }}b{\text{ are positive constants.}}\)</p>
<p>It is given that \({\text{P}}(X \geqslant 2) = 0.75\).</p>
</div>
<div class="specification">
<p>Eight independent observations of \(X\) are now taken and the random variable \(Y\) is the number of observations such that \(X \geqslant 2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(a = 32\) and \(b = \frac{1}{{12}}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{E}}(X)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{Var}}(X)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the median of \(X\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{E}}(Y)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{P}}(Y \geqslant 3)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(\int\limits_0^4 {\left( {\frac{{{x^2}}}{a} + b} \right){\text{d}}x = 1 \Rightarrow \left[ {\frac{{{x^3}}}{{3a}} + bx} \right]_0^4 = 1 \Rightarrow \frac{{64}}{{3a}} + 4b = 1} \) <strong><em>M1A1</em></strong></p>
<p>\(\int\limits_2^4 {\left( {\frac{{{x^2}}}{a} + b} \right){\text{d}}x = 0.75 \Rightarrow \frac{{56}}{{3a}} + 2b = 0.75} \) <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> \(\int\limits_0^2 {\left( {\frac{{{x^2}}}{a} + b} \right)} \,dx = 0.25 \Rightarrow \frac{8}{{3a}} + 2b = 0.25\) could be seen/used in place of either of the above equations.</p>
<p> </p>
<p>evidence of an attempt to solve simultaneously (or check given <em>a</em>,<em>b </em>values are consistent) <strong><em>M1</em></strong></p>
<p>\(a = 32,{\text{ }}b = \frac{1}{{12}}\) <strong><em>AG</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{E}}\left( X \right) = \int\limits_0^4 {x\left( {\frac{{{x^2}}}{{32}} + \frac{1}{{12}}} \right){\text{d}}x} \) <strong><em>(M1)</em></strong></p>
<p>\({\text{E}}(X) = \frac{8}{3}\,\,\,( = 2.67)\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{E}}\left( {{X^2}} \right) = \int\limits_0^4 {{x^2}\left( {\frac{{{x^2}}}{{32}} + \frac{1}{{12}}} \right){\text{d}}x} \) <strong><em>(M1)</em></strong></p>
<p>\({\text{Var}}(X) = {\text{E}}({X^2}) - {[{\text{E}}(X)]^2} = \frac{{16}}{{15}}\,\,\,( = 1.07)\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\int\limits_0^m {\left( {\frac{{{x^2}}}{{32}} + \frac{1}{{12}}} \right){\text{d}}x = 0.5} \) <strong><em>(M1)</em></strong></p>
<p>\(\frac{{{m^3}}}{{96}} + \frac{m}{{12}} = 0.5\,\,\,( \Rightarrow {m^3} + 8m - 48 = 0)\) <strong><em>(A1)</em></strong></p>
<p>\(m = 2.91\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(Y \sim B(8,{\text{ }}0.75)\) <strong><em>(M1)</em></strong></p>
<p>\({\text{E}}(Y) = 8 \times 0.75 = 6\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}(Y \geqslant 3) = 0.996\) <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>John likes to go sailing every day in July. To help him make a decision on whether it is safe to go sailing he classifies each day in July as windy or calm. Given that a day in July is calm, the probability that the next day is calm is 0.9. Given that a day in July is windy, the probability that the next day is calm is 0.3. The weather forecast for the 1st July predicts that the probability that it will be calm is 0.8.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Draw a tree diagram to represent this information for the first three days of July.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that the 3rd July is calm.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that the 1st July was calm given that the 3rd July is windy.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-09_om_18.23.32.png" alt="M17/5/MATHL/HP2/ENG/TZ2/05.a/M"> <strong><em>M1A2</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for 3 stage tree-diagram, <strong><em>A2 </em></strong>for 0.8, 0.9, 0.3 probabilities correctly placed.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(0.2 \times 0.7 \times 0.3 + 0.2 \times 0.3 \times 0.9 + 0.8 \times 0.1 \times 0.3 + 0.8 \times 0.9 \times 0.9 = 0.768\) <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}({\text{1st July is calm | 3rd July is windy)}} = \frac{{{\text{P}}({\text{1st July is calm and 3rd July is windy)}}}}{{{\text{P}}({\text{3rd July is windy)}}}}\) <strong><em>(M1)</em></strong></p>
<p>\( = \frac{{0.8 \times 0.1 \times 0.7 + 0.8 \times 0.9 \times 0.1}}{{1 - 0.768}}\)</p>
<p><strong>OR</strong>\(\,\,\,\,\,\)\(\frac{{0.8 \times 0.1 \times 0.7 + 0.8 \times 0.9 \times 0.1}}{{0.2 \times 0.7 \times 0.7 + 0.2 \times 0.3 \times 0.1 + 0.8 \times 0.1 \times 0.7 + 0.8 \times 0.9 \times 0.1}}\)</p>
<p><strong>OR</strong>\(\,\,\,\,\,\)\(\frac{{0.128}}{{0.232}}\) <strong><em>(</em></strong><strong><em>A1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for correct numerator, <strong><em>A1 </em></strong>for correct denominator.</p>
<p> </p>
<p>\( = 0.552\) <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the data set \(\{ k - 2,{\text{ }}k,{\text{ }}k + 1,{\text{ }}k + 4\} {\text{ , where }}k \in \mathbb{R}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the mean of this data set in terms of <em>k</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Each number in the above data set is now <strong>decreased</strong> by 3.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the mean of this <strong>new</strong> data set in terms of <em>k</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Use of \(\bar x = \frac{{\sum\limits_{i = 1}^4 {{x_i}} }}{n}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar x = \frac{{(k - 2) + k + (k + 1) + (k + 4)}}{4}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar x = \frac{{4k + 3}}{4}\,\,\,\,\,\left( { = k + \frac{3}{4}} \right)\) <strong><em>A1</em></strong> <strong><em>N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Either attempting to find the new mean or subtracting 3 from <strong>their</strong> \({\bar x}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar x = \frac{{4k + 3}}{4} - 3\,\,\,\,\,\left( { = \frac{{4k - 9}}{4},{\text{ }}k - \frac{9}{4}} \right)\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This was an easy question that was well done by most candidates. Careless arithmetic errors caused some candidates not to earn full marks. Only a few candidates realised that part (b) could be answered correctly by directly subtracting 3 from their answer to part (a). Most successful responses were obtained by redoing the calculation from part (a).</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Six balls numbered <span class="s1">1, 2, 2, 3, 3, 3 </span>are placed in a bag. Balls are taken one at a time from the bag at random and the number noted. Throughout the question a ball is always replaced before the next ball is taken.</p>
</div>
<div class="specification">
<p class="p1">Three balls are taken from the bag. Find the probability that</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A single ball is taken from the bag. Let \(X\) <span class="s1">denote the value shown on the ball.</span></p>
<p class="p2">Find \({\text{E}}(X)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">the total of the three numbers is <span class="s1">5</span>;</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">the median of the three numbers is <span class="s1">1</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Ten balls are taken from the bag. Find the probability that less than four of the balls are numbered <span class="s1">2</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the least number of balls that must be taken from the bag for the probability of taking out at least one ball numbered <span class="s1">2 </span>to be greater than <span class="s1">0.95</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Another bag also contains balls numbered <span class="s1">1 , 2 </span>or <span class="s1">3</span>.</p>
<p class="p1">Eight balls are to be taken from this bag at random. It is calculated that the expected number of balls numbered <span class="s1">1 </span>is <span class="s1">4.8 </span>, and the variance of the number of balls numbered 2 <span class="s2">is </span>1.5<span class="s2">.</span></p>
<p class="p1">Find the least possible number of balls numbered <span class="s1">3 </span>in this bag.</p>
<div class="marks">[8]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">\({\text{E}}(X) = 1 \times \frac{1}{6} + 2 \times \frac{2}{6} + 3 \times \frac{3}{6} = \frac{{14}}{6}{\text{ }}\left( { = \frac{7}{3} = 2.33} \right)\)Â <span class="Apple-converted-space">Â Â </span></span><strong><em>(M1)A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(3 \times {\text{P}}(113) + 3 \times {\text{P}}(122)\)Â <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\(3 \times \frac{1}{6} \times \frac{1}{6} \times \frac{1}{2} + 3 \times \frac{1}{6} \times \frac{1}{3} \times \frac{1}{3} = \frac{7}{{72}}{\text{ }}( = 0.0972)\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p3"><strong>Note: <span class="Apple-converted-space">Â Â </span></strong>Award <strong><em>M1 </em></strong>for attempt to find at least four of the cases.</p>
<p class="p3"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognising 111 as a possibility \(\left( {{\text{implied by }}\frac{1}{{216}}} \right)\)<span class="s1"> <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></span></p>
<p class="p1">recognising 112 and 113 as possibilities \(\left( {{\text{implied by }}\frac{2}{{216}}{\text{ and }}\frac{3}{{216}}} \right)\)<span class="s1"> <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></span></p>
<p class="p1">seeing the three arrangements of 112 and 113 <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\({\text{P}}(111) + 3 \times {\text{P}}(112) + 3 \times {\text{P}}(113)\)</p>
<p class="p1">\( = \frac{1}{{216}} + \frac{6}{{216}} + \frac{9}{{216}} = \frac{{16}}{{216}}{\text{ }}\left( { = \frac{2}{{27}} = 0.0741} \right)\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">let the number of twos be \(X,{\text{ }}X \sim B\left( {10,{\text{ }}\frac{1}{3}} \right)\) <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p2">\({\text{P}}(X < 4) = {\text{P}}(X \leqslant 3) = 0.559\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">let \(n\) </span>be the number of balls drawn</p>
<p class="p1">\({\text{P}}(X \geqslant 1) = 1 - {\text{P}}(X = 0)\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p1">\( = 1 - {\left( {\frac{2}{3}} \right)^n} > 0.95\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p1">\({\left( {\frac{2}{3}} \right)^n} > 0.05\)</p>
<p class="p1">\(n = 8\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(8{p_1} = 4.8 \Rightarrow {p_1} = \frac{3}{5}\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p1">\(8{p_2}(1 - {p_2}) = 1.5\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\(p_2^2 - {p_2} - 0.1875 = 0\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\({p_2} = \frac{1}{4}{\text{ }}\left( {{\text{or }}\frac{3}{4}} \right)\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">reject \(\frac{3}{4}\) as it gives a total greater than one</p>
<p class="p1">\({\text{P}}(1{\text{ or }}2) = \frac{{17}}{{20}}{\text{ or P}}(3) = \frac{3}{{20}}\) <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">recognising LCM as 20 so min total number is 20 <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">the least possible number of 3’s is 3 <span class="Apple-converted-space">  </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was generally well done, although many candidates lost their way after that.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Candidates had difficulty recognising all the different cases in part (b).</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Candidates had difficulty recognising all the different cases in part (b).</p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (c) and (d) should have been more standard questions, but many were unable to tackle them.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (c) and (d) should have been more standard questions, but many were unable to tackle them.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (e) was poorly answered in general.</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A continuous random variable \(T\) has probability density function \(f\) <span class="s1">defined by</span></p>
<p class="p2">\[f(t) = \left\{ {\begin{array}{*{20}{c}} {\frac{{t\left| {\sin 2t} \right|}}{\pi },}&{0 \leqslant t \leqslant \pi } \\ {0,}&{{\text{otherwise}}} \end{array}} \right.\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Sketch the graph of \(y = f(t)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use your sketch to find the mode of \(T\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the mean of \(T\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the variance of \(T\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that \(T\)Â lies between the mean and the mode.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space">  </span>Find \(\int_0^\pi  {f(t){\text{d}}t} \) <span class="s1">where \(0 \leqslant T \leqslant \frac{\pi }{2}\).</span></p>
<p class="p2">(ii) <span class="Apple-converted-space">Â Â </span>Hence verify that the lower quartile of \(T\) is \(\frac{\pi }{2}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-02-05_om_10.14.40.png" alt="M16/5/MATHL/HP2/ENG/TZ2/10.a/M"></p>
<p class="p1"><span class="s1">two enclosed regions (\(0 \leqslant t \leqslant \frac{\pi }{2}\) and \(\frac{\pi }{2} \leqslant t \leqslant \pi \)</span>) bounded by the curve and the \(t\)-axis <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></p>
<p class="p2">correct non-symmetrical shape for \(0 \leqslant t \leqslant \frac{\pi }{2}\) and</p>
<p class="p2"><span class="Apple-converted-space">\(\frac{\pi }{2} < {\text{mode of }}T < \pi {\text{ clearly apparent}}\) Â Â </span><span class="s2"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({\text{mode}} = 2.46\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({\text{E}}(T) = \frac{1}{\pi }\int_0^\pi  {{t^2}\left| {\sin 2t} \right|{\text{d}}t} \)   </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\( = 2.04\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>EITHER</strong></p>
<p class="p1"><span class="s1">\({\text{Var}}(T) = \int_0^\pi  {(t - } 2.03788 \ldots {)^2}\left( {\frac{{t\left| {\sin 2t} \right|}}{\pi }} \right){\text{d}}t\) <span class="Apple-converted-space">  </span></span><strong><em>(M1)(A1)</em></strong></p>
<p class="p1"><strong>OR</strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{Var}}(T) = \int_0^\pi  {{t^2}} \left( {\frac{{t\left| {\sin 2t} \right|}}{\pi }} \right){\text{d}}t - {(2.03788 \ldots )^2}\)    </span></p>
<p class="p1"><strong><em>(M1)(A1)</em></strong></p>
<p class="p1"><strong>THEN</strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{Var}}(T) = 0.516\) Â Â </span><span class="s2"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(\frac{1}{\pi }\int_{{\text{2.03788}} \ldots }^{{\text{2.456590}} \ldots } {t\left| {\sin 2t} \right|{\text{d}}t = {\text{0.285}}} \) Â Â </span><strong><em>(M1)A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space">Â Â </span>attempting integration by parts <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="s1">\((u = t,{\text{ d}}u = {\text{d}}t,{\text{ d}}v = \sin 2t{\text{ d}}t\) </span>and \(v =  - \frac{1}{2}\cos 2t)\)</p>
<p class="p2"><span class="Apple-converted-space">\(\frac{1}{\pi }\left[ {t\left( { - \frac{1}{2}\cos 2t} \right)} \right]_0^r - \frac{1}{\pi }\int_0^r {\left( { - \frac{1}{2}\cos 2t} \right){\text{d}}t} \) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="s1"><strong>Note: <span class="Apple-converted-space">Â Â </span></strong>Award <strong><em>A1 </em></strong></span>if the limits are not included.</p>
<p class="p2"><span class="Apple-converted-space">\( = \frac{{\sin 2T}}{{4\pi }} - \frac{{T\cos 2T}}{{2\pi }}\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2">(ii) <span class="Apple-converted-space">Â Â \(\frac{{\sin \pi }}{{4\pi }} - \frac{{\frac{\pi }{2}\cos \pi }}{{2\pi }} = \frac{1}{4}\)</span>Â <span class="Apple-converted-space">Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s2">as \({\text{P}}\left( {0 \leqslant T \leqslant \frac{\pi }{2}} \right) = \frac{1}{4}\)Â </span>(or equivalent), then the lower quartile of \(T\) <span class="s2">is \(\frac{\pi }{2}\)Â <span class="Apple-converted-space">Â Â </span></span><strong><em>R1AG</em></strong></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was generally accessible to the large majority of candidates. A substantial number of candidates were able to neatly and accurately sketch a non-symmetric bimodal continuous probability density function and to calculate its mean, mode and variance. Quite a few candidates unfortunately attempted this question with their GDC set in degrees.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was generally accessible to the large majority of candidates. A substantial number of candidates were able to neatly and accurately sketch a non-symmetric bimodal continuous probability density function and to calculate its mean, mode and variance. Quite a few candidates unfortunately attempted this question with their GDC set in degrees.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was generally accessible to the large majority of candidates. A substantial number of candidates were able to neatly and accurately sketch a non-symmetric bimodal continuous probability density function and to calculate its mean, mode and variance. Quite a few candidates unfortunately attempted this question with their GDC set in degrees.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was generally accessible to the large majority of candidates. A substantial number of candidates were able to neatly and accurately sketch a non-symmetric bimodal continuous probability density function and to calculate its mean, mode and variance. Quite a few candidates unfortunately attempted this question with their GDC set in degrees.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was generally accessible to the large majority of candidates. A substantial number of candidates were able to neatly and accurately sketch a non-symmetric bimodal continuous probability density function and to calculate its mean, mode and variance. Quite a few candidates unfortunately attempted this question with their GDC set in degrees.</p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was generally accessible to the large majority of candidates. A substantial number of candidates were able to neatly and accurately sketch a non-symmetric bimodal continuous probability density function and to calculate its mean, mode and variance. Quite a few candidates unfortunately attempted this question with their GDC set in degrees.</p>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>The continuous random variable <em>X</em> has probability density function \(f\) given by</p>
<p>\[f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}<br> {3ax}&,&{0 \leqslant x < 0.5} \\ <br> {a\left( {2 - x} \right)}&,&{0.5 \leqslant x < 2} \\ <br> 0&,&{{\text{otherwise}}} <br>\end{array}} \right.\]</p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(a = \frac{2}{3}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{P}}\left( {X < 1} \right)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \({\text{P}}\left( {s < X < 0.8} \right) = 2 \times {\text{P}}\left( {2s < X < 0.8} \right)\), and that 0.25 < <em>s</em> < 0.4 , find the value of <em>s</em>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>Â </p>
<p>\(a\left[ {\int_0^{0.5} {3x\,{\text{d}}x}Â + \int_{0.5}^2 {\left( {2 - x} \right)} \,{\text{d}}x} \right] = 1\)Â Â Â <em><strong>M1</strong></em></p>
<p><strong>Note</strong>: Award the <em><strong>M1</strong></em> for the total integral equalling 1, or equivalent.</p>
<p>\(a\left( {\frac{3}{2}} \right) = 1\)Â Â Â <em><strong>(M1)A1</strong></em></p>
<p>\(a = \frac{2}{3}\)Â Â Â <em><strong>AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>\(\int_0^{0.5} {2x\,{\text{d}}x}Â + \frac{2}{3}\int_{0.5}^1 {\left( {2 - x} \right)} \,{\text{d}}x\)Â Â <em><strong>Â (M1)(A1)</strong></em></p>
<p>\( = \frac{2}{3}\)Â Â Â <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p>\(\frac{2}{3}\int_1^2 {\left( {2 - x} \right)} \,{\text{d}}x = \frac{1}{3}\)Â Â Â <em><strong>(M1)</strong></em></p>
<p>so \({\text{P}}\left( {X < 1} \right) = \frac{2}{3}\)    <em><strong>(M1)A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}\left( {s < X < 0.8} \right) = \int_s^{0.5} {2x\,{\text{d}}x}Â + \frac{2}{3}\int_{0.5}^{0.8} {\left( {2 - x} \right)} \,{\text{d}}x\)Â Â Â <em><strong>M1A1</strong></em></p>
<p>\( = \left[ {{x^2}} \right]_s^{0.5} + 0.27\)</p>
<p>\(0.25 - {s^2} + 0.27\)Â Â <strong>Â <em>(A1)</em></strong></p>
<p>\({\text{P}}\left( {2s < X < 0.8} \right) = \frac{2}{3}\int_{2s}^{0.8} {\left( {2 - x} \right)} \,{\text{d}}x\)Â Â Â <em><strong>A1</strong></em></p>
<p>\( = \frac{2}{3}\left[ {2x - \frac{{{x^2}}}{2}} \right]_{2s}^{0.8}\)</p>
<p>\(\frac{2}{3}\left( {1.28 - \left( {4s - 2{s^2}} \right)} \right)\)</p>
<p>equating</p>
<p>\(0.25 - {s^2} + 0.27 = \frac{4}{3}\left( {1.28 - \left( {4s - 2{s^2}} \right)} \right)\)Â Â Â <em><strong>(A1)</strong></em></p>
<p>attempt to solve for <em>s</em>Â Â Â <em><strong>(M1)</strong></em></p>
<p><em>s</em>Â = 0.274Â Â Â <em><strong>A1</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Students sign up at a desk for an activity during the course of an afternoon. The arrival of each student is independent of the arrival of any other student and the number of students arriving per hour can be modelled as a Poisson distribution with a mean of \(\lambda \)<span class="s1">.</span></p>
<p class="p2">The desk is open for 4 hours. If exactly 5 people arrive to sign up for the activity during that time find the probability that exactly 3 <span class="s2">of them arrived during the first hour.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1"><span class="s1">P(3 </span>in the first hour) \( = \frac{{{\lambda ^3}{e^{ - \lambda }}}}{{3!}}\) <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></p>
<p class="p1">number to arrive in the four hours follows \(Po(4\lambda )\) <span class="Apple-converted-space">Â Â </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="s1">P(5 </span>arrive in total) \( = \frac{{{{(4\lambda )}^5}{e^{ - 4\lambda }}}}{{5!}}\) <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="s1">attempt to find P(2 </span>arrive in the next three hours) <span class="Apple-converted-space">Â Â </span><strong><em>M1</em></strong></p>
<p class="p1">\( = \frac{{{{(3\lambda )}^2}{e^{ - 3\lambda }}}}{{2!}}\) <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></p>
<p class="p1">use of conditional probability formula <span class="Apple-converted-space">Â Â </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="s1">P(3 in the first hour given 5 </span>in total) \( = \frac{{\frac{{{\lambda ^3}{e^{ - \lambda }}}}{{3!}} \times \frac{{{{(3\lambda )}^2}{e^{ - 3\lambda }}}}{{2!}}}}{{\frac{{{{(4\lambda )}^5}{e^{ - 4\lambda }}}}{{5!}}}}\) <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></p>
<p class="p1">\(\frac{{\left( {\frac{9}{{2!3!}}} \right)}}{{\left( {\frac{{{4^5}}}{{5!}}} \right)}} = \frac{{45}}{{512}} = 0.0879\) <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[8 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">A more difficult question, but it was still surprising how many candidates were unable to make a good start with it. Many were using \(\lambda = \frac{5}{4}\) and consequently unable to progress very far. Many students failed to recognise that a conditional probability should be used.</p>
</div>
<br><hr><br><div class="specification">
<p>Packets of biscuits are produced by a machine. The weights \(X\), in grams, of packets of biscuits can be modelled by a normal distribution where \(X \sim {\text{N}}(\mu ,{\text{ }}{\sigma ^2})\). A packet of biscuits is considered to be underweight if it weighs less than 250 grams.</p>
</div>
<div class="specification">
<p>The manufacturer makes the decision that the probability that a packet is underweight should be 0.002. To do this \(\mu \) is increased and \(\sigma \) remains unchanged.</p>
</div>
<div class="specification">
<p>The manufacturer is happy with the decision that the probability that a packet is underweight should be 0.002, but is unhappy with the way in which this was achieved. The machine is now adjusted to reduce \(\sigma \) and return \(\mu \) to 253.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that \(\mu = 253\) and \(\sigma = 1.5\) find the probability that a randomly chosen packet of biscuits is underweight.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the new value of \(\mu \) giving your answer correct to two decimal places.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the new value of \(\sigma \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}(X < 250) = 0.0228\) <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\frac{{250 - \mu }}{{1.5}} = - 2.878 \ldots \) <strong><em>(M1)(A1)</em></strong></p>
<p>\( \Rightarrow \mu = 254.32\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Only award <strong><em>A1 </em></strong>here if the correct 2dp answer is seen. Award <strong><em>M0 </em></strong>for use of \({1.5^2}\).</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\frac{{250 - 253}}{\sigma } = - 2.878 \ldots \) <strong><em>(A1)</em></strong></p>
<p>\( \Rightarrow \sigma = 1.04\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The number of cats visiting Helena’s garden each week follows a Poisson distribution with mean \(\lambda = 0.6\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) in a particular week no cats will visit Helena’s garden;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) in a particular week at least three cats will visit Helena’s garden;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) over a four-week period no more than five cats in total will visit Helena’s garden;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) over a twelve-week period there will be exactly four weeks in which at least one cat will visit Helena’s garden.</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A continuous random variable \(X\) has probability distribution function \(f\) given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \(f(x) = k\ln x\) \(1 \leqslant x \leqslant 3\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \(f(x) = 0\) otherwise</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the value of \(k\) to six decimal places.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the value of \({\text{E}}(X)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) State the mode of \(X\)<em>.</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) Find the median of \(X\)<em>.</em></span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(X \sim {\text{Po(0.6)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 0) = 0.549{\text{ }}\left( { = {{\text{e}}^{ - 0.6}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\text{P}}(X \geqslant 3) = 1 - {\text{P}}(X \leqslant 2)\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 1 - \left( {{{\text{e}}^{ - 0.6}} + {{\text{e}}^{ - 0.6}} \times 0.6 + {{\text{e}}^{ - 0.6}} \times \frac{{{{0.6}^2}}}{2}} \right)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.0231\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \(Y \sim {\text{Po(2.4)}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Y \leqslant 5) = 0.964\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) \(Z \sim {\text{B(12, 0.451}} \ldots )\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1 </em></strong>for recognising binomial and <strong><em>A1 </em></strong>for using correct parameters.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Z = 4) = 0.169\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(k\int_1^3 {\ln x{\text{d}}x = 1} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\((k \times 1.2958 \ldots = 1)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(k = 0.771702\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\text{E}}(X) = \int_1^3 {kx\ln x{\text{d}}x} \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to evaluate their integral <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 2.27\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \(x = 3\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) \(\int_1^m {k\ln x{\text{d}}x = 0.5} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(k[x\ln x - x]_1^m = 0.5\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to solve for <em>m</em> <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(m = 2.34\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[9 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were generally well done by a large proportion of candidates. In part (a) (ii), some candidates used an incorrect inequality (e.g. \({\text{P}}(X \geqslant 3) = 1 - {\text{P}}(X \leqslant 3)\)) while in (a) (iii) some candidates did not use \(\mu = 2.4\). In part (a) (iv), a number of candidates either did not realise that they needed to consider a binomial random variable or did so using incorrect parameters.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were generally well done by a large proportion of candidates.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In (b) (i), some candidates gave their value of <em>k</em> correct to three significant figures rather than correct to six decimal places. In parts (b) (i), (ii) and (iv), a large number of candidates unnecessarily used integration by parts. In part (b) (iii), a number of candidates thought the mode of <em>X</em> was \(f(3)\) rather than \(x = 3\). In part (b) (iv), a number of candidates did not consider the domain of <em>f</em> when attempting to find the median or checking their solution.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In a factory producing glasses, the weights of glasses are known to have a mean of 160 grams. It is also known that the interquartile range of the weights of glasses is 28 grams. Assuming the weights of glasses to be normally distributed, find the standard deviation of the weights of glasses.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">weight of glass = <em>X</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{N}}(160,{\text{ }}{\sigma ^2})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X < 160 + 14) = {\text{P}}(X < 174) = 0.75\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> \({\text{P}}(X < 160 - 14) = {\text{P}}(X < 146) = 0.25\) can also be used.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}\left( {Z < \frac{{14}}{\sigma }} \right) = 0.75\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{14}}{\sigma } = 0.6745 \ldots \) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sigma = 20.8\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Of those students able to start the question, there were good solutions seen. Most students could have made better use of the GDC on this question.</span></p>
</div>
<br><hr><br><div class="specification">
<p>The age, <em>L</em>, in years, of a wolf can be modelled by the normal distribution <em>L</em> ~ N(8, 5).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a wolf selected at random is at least 5 years old.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Eight wolves are independently selected at random and their ages recorded.</p>
<p>Find the probability that more than six of these wolves are at least 5 years old.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>P(<em>L</em> ≥ 5) = 0.910   <em><strong>(M1)A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>X</em> is the number of wolves found to be at least 5 years old recognising binomial distribution   <em><strong>M1</strong></em></p>
<p><em>X</em> ~ B(8, 0.910…)</p>
<p>P(<em>X</em> > 6) = 1 − P(<em>X</em> ≤ 6)   <em><strong>(M1)</strong></em></p>
<p>=Â 0.843Â Â Â Â <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for finding P(<em>X</em> ≥ 6).</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The distance travelled by students to attend Gauss College is modelled by a normal distribution with mean 6 km and standard deviation 1.5 km.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the probability that the distance travelled to Gauss College by a </span><span style="font-family: 'times new roman', times; font-size: medium;">randomly selected student is between 4.8 km and 7.5 km.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) 15 % of students travel less than <em>d</em> km to attend Gauss College. Find the </span><span style="font-family: 'times new roman', times; font-size: medium;">value of <em>d</em>.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">At Euler College, the distance travelled by students to attend their school is modelled by a normal distribution with mean \(\mu \) km and standard deviation \(\sigma \) km.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If 10 % of students travel more than 8 km and 5 % of students travel less than 2 km, find the value of \(\mu \) and of \(\sigma \) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The number of telephone calls, <em>T</em>, received by Euler College each minute can be modelled by a Poisson distribution with a mean of 3.5.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the probability that at least three telephone calls are received by Euler College in <strong>each</strong> of two successive one-minute intervals.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the probability that Euler College receives 15 telephone calls during a randomly selected five-minute interval.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\text{P}}(4.8 < X < 7.5) = {\text{P}}( - 0.8 < Z < 1)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.629 <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept \({\text{P}}(4.8 \leqslant X \leqslant 7.5) = {\text{P}}( - 0.8 \leqslant Z \leqslant 1)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Stating \({\text{P}}(X < d) = 0.15\) <strong>or</strong> sketching an appropriately labelled diagram. <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{d - 6}}{{1.5}} = - 1.0364…\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>d</em> = (−1.0364...)(1.5) + 6 <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 4.45 (km) <strong><em>A1</em></strong> <strong><em>N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Stating <strong>both</strong> \({\text{P}}(X > 8) = 0.1\) and \({\text{P}}(X < 2) = 0.05\) <strong>or</strong> sketching an appropriately labelled diagram. <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Setting up two equations in \(\mu \) and \(\sigma \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">8 = \(\mu \) + (1.281…)\(\sigma \) <strong>and</strong> 2 = \(\mu \) − (1.644…)\(\sigma \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Attempting to solve for \(\mu \) and \(\sigma \) (including by graphical means) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sigma \) = 2.05 (km) <strong>and</strong> \(\mu \) = 5.37 (km) <strong><em>A1A1 N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept \(\mu \) = 5.36, 5.38 .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Use of the Poisson distribution in an inequality. <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(T \geqslant 3) = 1 - {\text{P}}(T \leqslant 2)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.679... <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Required probability is \({(0.679…)^2} = 0.461\) <strong><em>M1A1</em></strong> <strong><em>N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Allow <strong><em>FT</em></strong> for their value of \({\text{P}}(T \geqslant 3)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(\tau \sim {\text{Po(17.5)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(\tau = 15) = \frac{{{{\text{e}}^{ - 17.5}}{{(17.5)}^{15}}}}{{15!}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.0849 <strong><em>A1</em></strong> <strong><em>N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally well done despite a large proportion of candidates being awarded an accuracy penalty. Candidates found part (a) (i) to be quite straightforward and was generally done very well. In part (a) (ii), a number of candidates used \(\frac{{d - 6}}{{1.5}} = 1.0364…\) instead of \(\frac{{d - 6}}{{1.5}} = - 1.0364…\) . In part (b), a pleasingly high number of candidates were able to set up and solve a pair of simultaneous linear equations to correctly find the values of \(\mu \) and \(\sigma \). Some candidates prematurely rounded intermediate results. In part (c), a number of candidates were unable to express a correct Poisson inequality. Common errors included stating \({\text{P}}(T \geqslant 3) = 1 - {\text{P}}(T \leqslant 3)\) and using \(\mu = 7\).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally well done despite a large proportion of candidates being awarded an accuracy penalty. Candidates found part (a) (i) to be quite straightforward and was generally done very well. In part (a) (ii), a number of candidates used \(\frac{{d - 6}}{{1.5}} = 1.0364…\) instead of \(\frac{{d - 6}}{{1.5}} = - 1.0364…\) . In part (b), a pleasingly high number of candidates were able to set up and solve a pair of simultaneous linear equations to correctly find the values of \(\mu \) and \(\sigma \). Some candidates prematurely rounded intermediate results. In part (c), a number of candidates were unable to express a correct Poisson inequality. Common errors included stating \({\text{P}}(T \geqslant 3) = 1 - {\text{P}}(T \leqslant 3)\) and using \(\mu = 7\).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was generally well done despite a large proportion of candidates being awarded an accuracy penalty. Candidates found part (a) (i) to be quite straightforward and was generally done very well. In part (a) (ii), a number of candidates used \(\frac{{d - 6}}{{1.5}} = 1.0364…\) instead of \(\frac{{d - 6}}{{1.5}} = - 1.0364…\) . In part (b), a pleasingly high number of candidates were able to set up and solve a pair of simultaneous linear equations to correctly find the values of \(\mu \) and \(\sigma \). Some candidates prematurely rounded intermediate results. In part (c), a number of candidates were unable to express a correct Poisson inequality. Common errors included stating \({\text{P}}(T \geqslant 3) = 1 - {\text{P}}(T \leqslant 3)\) and using \(\mu = 7\).</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A random variable \(X\) has probability density function</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ \begin{array}{r}ax + b,\\0,\end{array} \right.\begin{array}{*{20}{c}}{2 \le x \le 3}\\{{\rm{ otherwise}}}\end{array},a,b \in \mathbb{R}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(5a + 2b = 2\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \({\text{E}}(X) = \mu \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Show that \(a = 12\mu - 30\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Find a similar expression for <em>b </em>in terms of \(\mu \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let the median of the distribution be 2.3.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) Find the value of \(\mu \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Find the value of the standard deviation of <em>X</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\int_2^3 {(ax + b){\text{d}}x{\text{ }}( = 1)} \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left[ {\frac{1}{2}a{x^2} + bx} \right]_2^3{\text{ }}( = 1)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{5}{2}a + b = 1\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(5a + 2b = 2\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) \(\int_2^3 {\left( {a{x^2} + bx} \right){\text{d}}x{\text{ }}( = \mu )} \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(\left[ {\frac{1}{3}a{x^3} + \frac{1}{2}b{x^2}} \right]_2^3{\text{ }}( = \mu )\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(\frac{{19}}{3}a + \frac{5}{2}b = \mu \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> eliminating <em>b</em> <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em> eg</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(\frac{{19}}{3}a + \frac{5}{2}\left( {1 - \frac{5}{2}a} \right) = \mu \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(\frac{1}{{12}}a + \frac{5}{2} = \mu \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(a = 12\mu - 30\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Elimination of <em>b </em>could be at different stages.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) \(b = 1 - \frac{5}{2}(12\mu - 30)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = 76 - 30\mu \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> This solution may be seen in part (i).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) \(\int_2^{2.3} {(ax + b){\text{d}}x{\text{ }}( = 0.5)} \) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(\left[ {\frac{1}{2}a{x^2} + bx} \right]_2^{2.3}{\text{ }}( = 0.5)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(0.645a + 0.3b{\text{ }}( = 0.5)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(0.645(12\mu - 30) + 0.3(76 - 30\mu ) = 0.5\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(\mu = 2.34{\text{ }}\left( { = \frac{{295}}{{126}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) \({\text{E}}\left( {{X^2}} \right) = \int_2^3 {{x^2}(ax + b){\text{d}}x} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(a = 12\mu - 30 = - \frac{{40}}{{21}},{\text{ }}b = 76 - 30\mu = \frac{{121}}{{21}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \({\text{E}}\left( {{X^2}} \right) = \int_2^3 {{x^2}\left( { - \frac{{40}}{{21}}x + \frac{{121}}{{21}}} \right){\text{d}}x = 5.539 \ldots {\text{ }}\left( { = \frac{{349}}{{63}}} \right)} \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \({\text{Var}}(X) = 5.539{\text{K}} - {(2.341{\text{K}})^2} = 0.05813 \ldots \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(\sigma = 0.241\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [21 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A market stall sells apples, pears and plums.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weights of the apples are normally distributed with a mean of 200 grams and a standard deviation of 25 grams.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Given that there are 450 apples on the stall, what is the expected number of apples with a weight of more than 225 grams?</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Given that 70 % of the apples weigh less than <em>m </em>grams, find the value of <em>m </em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weights of the pears are normally distributed with a mean of ∝ grams and a standard deviation of \(\sigma \)<span style="font: 12.5px Helvetica;"> </span>grams. Given that 8 % of these pears have a weight of more than 270 grams and 15 % have a weight less than 250 grams, find ∝ and \(\sigma \)<span style="font: 12.5px Helvetica;"> </span>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weights of the plums are normally distributed with a mean of 80 grams and a standard deviation of 4 grams. 5 plums are chosen at random. What is the probability that exactly 3 of them weigh more than 82 grams?</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\text{P}}(X > 225) = 0.158...\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">expected number \( = 450 \times 0.158... = 71.4\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\text{P}}(X < m) = 0.7\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow m = 213{\text{ (grams)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[5 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{270 - \mu }}{\sigma } = 1.40...\) <strong> <em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{250 - \mu }}{\sigma } = - 1.03...\) <strong> <em>A1</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"> </strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>These could be seen in graphical form.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">solving simultaneously <strong><em>(M1)<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>\(\mu = 258,{\text{ }}\sigma = 8.19\) </em><strong><em> A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[6 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{N}}({80,4^2})\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X > 82) = 0.3085...\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">recognition of the use of binomial distribution. <strong><em>(M1)<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{B}}(5,\,0.3085...)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 3) = 0.140\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was an accessible question for most students with many wholly correct answers seen. In part (b) a few candidates struggled to find the correct values from the calculator and in part (c) a small minority did not see the need to treat it as a binomial distribution.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was an accessible question for most students with many wholly correct answers seen. In part (b) a few candidates struggled to find the correct values from the calculator and in part (c) a small minority did not see the need to treat it as a binomial distribution.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This was an accessible question for most students with many wholly correct answers seen. In part (b) a few candidates struggled to find the correct values from the calculator and in part (c) a small minority did not see the need to treat it as a binomial distribution.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A ferry carries cars across a river. There is a fixed time of <em>T</em> minutes between crossings. The arrival of cars at the crossing can be assumed to follow a Poisson distribution with a mean of one car every four minutes. Let <em>X</em> denote the number of cars that arrive in <em>T</em> minutes.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find <em>T</em>, to the nearest minute, if \({\text{P}}(X \leqslant 3) = 0.6\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">It is now decided that the time between crossings, <em>T</em>, will be 10 minutes. The ferry can carry a maximum of three cars on each trip.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">One day all the cars waiting at 13:00 get on the ferry. Find the probability that all the cars that arrive in the next 20 minutes will get on either the 13:10 or the 13:20 ferry.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{Po(0.25T)}}\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Attempt to solve \({\text{P}}(X \leqslant 3) = 0.6\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(T = 12.8453 \ldots = 13{\text{ (minutes)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1M1A0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> if </span><em style="font-family: 'times new roman', times; font-size: medium;">T</em><span style="font-family: 'times new roman', times; font-size: medium;"> found correctly but not stated to the nearest minute.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">let \({X_1}\) be the number of cars that arrive during the first interval and \({X_2}\) be the number arriving during the second.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({X_1}\) and \({X_2}\) are Po(2.5) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">P (all get on) \( = {\text{P}}({X_1} \leqslant 3) \times {\text{P}}({X_2} \leqslant 3) + {\text{P}}({X_1} = 4) \times {\text{P}}({X_2} \leqslant 2)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( + {\text{P}}({X_1} = 5) \times {\text{P}}({X_2} \leqslant 1) + {\text{P}}({X_1} = 6) \times {\text{P}}({X_2} = 0)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.573922 \ldots + 0.072654 \ldots + 0.019192 \ldots + 0.002285 \ldots \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.668{\text{ }}(053 \ldots )\) <strong><em>A1 </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were some good answers to part (a), although poor calculator use frequently let down the candidates.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Very few candidates were able to access part (b).</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A random variable \(X\) is normally distributed with mean \(\mu \) and standard deviation \(\sigma \), such that \({\text{P}}(X < 30.31) = 0.1180\) and \({\text{P}}(X > 42.52) = 0.3060\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\mu \) and \(\sigma \).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{P}}\left( {\left| {X - \mu } \right| < 1.2\sigma } \right)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(X < 42.52) = 0.6940\) Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p1">either \({\text{P}}\left( {Z < \frac{{30.31 - \mu }}{\sigma }} \right) = 0.1180{\text{ or P}}\left( {Z < \frac{{42.52 - \mu }}{\sigma }} \right) = 0.6940\)Â <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(\frac{{30.31 - \mu }}{\sigma } = \underbrace {{\Phi ^{ - 1}}(0.1180)}_{ - 1.1850 \ldots }\) Â Â </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(\frac{{42.52 - \mu }}{\sigma } = \underbrace {{\Phi ^{ - 1}}(0.6940)}_{0.5072 \ldots }\) Â Â </span><strong><em>(A1)</em></strong></p>
<p class="p1">attempting to solve simultaneously <span class="Apple-converted-space">Â Â </span><strong><em>(M1) </em></strong></p>
<p class="p1">\(\mu  = 38.9\) and \(\sigma  = 7.22\) <span class="Apple-converted-space">  </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P}}(\mu  - 1.2\sigma  < X < \mu  + 1.2\sigma )\) (or equivalent <em>eg</em>. \(2{\text{P}}(\mu  < X < \mu  + 1.2\sigma )\)) <span class="Apple-converted-space">  </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = 0.770\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>(M1)A1 </em></strong>for \({\text{P}}( - 1.2 < Z < 1.2) = 0.770\).</p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The number of vehicles passing a particular junction can be modelled using the Poisson distribution. Vehicles pass the junction at an average rate of 300 per hour.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that no vehicles pass in a given minute.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the expected number of vehicles which pass in a given two minute period.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">Find the probability that more than this expected number actually pass in a given two minute period.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(m = \frac{{300}}{{60}} = 5\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 0) = 0.00674\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">or \({{\text{e}}^{ - 5}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(X) = 5 \times 2 = 10\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X > 10) = 1 - {\text{P}}(X \leqslant 10)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.417 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were answered successfully by many candidates. Some candidates had difficulty obtaining the correct inequality in (c).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were answered successfully by many candidates. Some candidates had difficulty obtaining the correct inequality in (c).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were answered successfully by many candidates. Some candidates had difficulty obtaining the correct inequality in (c).</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The finishing times in a marathon race follow a normal distribution with mean <span class="s1">210 </span>minutes and standard deviation <span class="s1">22 </span>minutes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that a runner finishes the race in under three hours.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The fastest \(90\% \) of the finishers receive a certificate.</p>
<p class="p1">Find the time, below which a competitor has to complete the race, in order to gain a certificate.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(X \sim N(210,{\text{ }}{22^2})\)</p>
<p>\({\text{P}}(X < 180) = 0.0863\) <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P}}(X < T) = 0.9 \Rightarrow T = 238{\text{ (mins)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[2 marks]</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>Total [5 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was well done with many candidates obtaining full marks. On the whole, but quite a number misunderstood what was required in part (b) and 182 minutes was a repeated incorrect answer. It was disappointing that candidates have not noticed that this answer was clearly too small showing that candidates had not appreciated the context of the question.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was well done with many candidates obtaining full marks. On the whole, but quite a number misunderstood what was required in part (b) and 182 minutes was a repeated incorrect answer. It was disappointing that candidates have not noticed that this answer was clearly too small showing that candidates had not appreciated the context of the question.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The continuous random variable \(X\) has the probability distribution function \(f(x) = A\sin \left( {\ln (x)} \right),{\text{ }}1 \le x \le 5\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(A\) to three decimal places.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the mode of \(X\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value \({\text{P}}(X \le 3|X \ge 2)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(A\int_1^5 {\sin (\ln x){\text{d}}x = 1} \) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\(A = 0.323{\text{ (3 dp only)}}\) <strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">either a graphical approach or \(f'(x) = \frac{{\cos (\ln x)}}{x} = 0\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\(x = 4.81\;\;\;\left( { = {{\text{e}}^{\frac{\pi }{2}}}} \right)\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Do not award <strong><em>A1FT </em></strong>for a candidate working in degrees.</p>
<p class="p1"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P}}(X \le 3|X \ge 2) = \frac{{{\text{P}}(2 \le X \le 3)}}{{{\text{P}}(X \ge 2)}}\;\;\;\left( { = \frac{{\int_2^3 {\sin \left( {\ln (x)} \right){\text{d}}x} }}{{\int_2^5 {\sin \left( {\ln (x)} \right){\text{d}}x} }}} \right)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\( = 0.288\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Do not award <strong><em>A1FT </em></strong>for a candidate working in degrees.</p>
<p class="p1"><em><strong>[2 marks]</strong></em></p>
<p class="p1"><em><strong>Total [6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Josie has three ways of getting to school. <span class="s1">\(30\% \) </span>of the time she travels by car, <span class="s1">\(20\% \) </span>of the time she rides her bicycle and <span class="s1">\(50\% \) </span>of the time she walks.</p>
<p class="p1">When travelling by car, Josie is late <span class="s1">\(5\% \) </span>of the time. When riding her bicycle she is late <span class="s1">\(10\% \) </span>of the time. When walking she is late <span class="s1">\(25\% \) </span>of the time. Given that she was on time, find the probability that she rides her bicycle.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1"><strong>EITHER</strong></p>
<p class="p1"><strong><em><img src="images/Schermafbeelding_2016-01-29_om_10.48.05.png" alt> M1A1A1</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>M1 </em></strong>for a two-level tree diagram, <strong><em>A1 </em></strong>for correct first level probabilities, and <strong><em>A1 </em></strong>for correct second level probabilities.</p>
<p class="p2"> </p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">\({\text{P}}(B|L') = \frac{{{\text{P}}(L'|B){\text{P}}(B)}}{{{\text{P}}(L'|B){\text{P}}(B) + {\text{P}}(L'|C){\text{P}}(C) + {\text{P}}(L'|W){\text{P}}(Q)}}\;\;\;\left( { = \frac{{{\text{P}}(B \cap L'}}{{{\text{P}}(L')}}} \right)\) <span class="Apple-converted-space"> </span><strong><em>(M1)(A1)(A1)</em></strong></p>
<p class="p1"><strong>THEN</strong></p>
<p class="p1">\({\text{P}}(B|L') = \frac{{0.9 \times 0.2}}{{0.9 \times 0.2 + 0.95 \times 0.3 + 0.75 \times 0.5}}\;\;\;\left( { = \frac{{0.18}}{{0.84}}} \right)\) <span class="Apple-converted-space"> </span><strong><em>M1A1</em></strong></p>
<p class="p1">\( = 0.214\;\;\;\left( { = \frac{3}{{14}}} \right)\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A continuous random variable <em>X</em> has probability density function</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {12{x^2}(1 - x),}&{{\text{for }}0 \leqslant x \leqslant 1,} \\ <br> {0,}&{{\text{otherwise}}{\text{.}}} <br>\end{array}} \right.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that <em>X</em> lies between the mean and the mode.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Attempting to find the mode graphically or by using \(f'(x) = 12x(2 - 3x)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Mode}} = \frac{2}{3}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use of \({\text{E}}(X) = \int_0^1 {xf(x){\text{d}}x} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(X) = \frac{3}{5}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_{\frac{3}{5}}^{\frac{2}{3}} {f(x){\text{d}}x = 0.117\,\,\,\,\,\left( { = \frac{{1981}}{{16\,{\text{875}}}}} \right)} \) <strong><em>M1A1 N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A significant number of candidates attempted to find the mode and the mean using calculus when it could be argued that these quantities could be found more efficiently with a GDC. </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A significant proportion of candidates demonstrated a lack of understanding of the definitions governing the mean, mode and median of a continuous probability density function. A significant number of candidates attempted to calculate the median instead of either the mean or the mode. A number of candidates prematurely rounded their value for the mode i.e. subsequently using 0.7 for example rather than using the exact value of \(\frac{2}{3}\). A few candidates offered negative probability values or probabilities greater than one.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The data of the goals scored by players in a football club during a season are given in the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-29_om_10.01.43.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that the mean number of goals scored per player is <span class="s1">\(1.95\) </span>, find the value of \(k\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">It is discovered that there is a mistake in the data and that the top scorer, who scored <span class="s1">22 </span>goals, has not been included in the table.</p>
<p class="p1">(i) Find the correct mean number of goals scored per player.</p>
<p class="p1">(ii) Find the correct standard deviation of the number of goals scored per player.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(\frac{{0 \bullet 4 + 1 \bullet k + 2 \bullet 3 + 3 \bullet 2 + 4 \bullet 3 + 8 \bullet 1}}{{13 + k}} = 1.95\;\;\;\left( {\frac{{k + 32}}{{k + 13}} = 1.95} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2">attempting to solve for \(k\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\(k = 7\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(\frac{{7 + 32 + 22}}{{7 + 13 + 1}} = 2.90\;\;\;\left( { = \frac{{61}}{{21}}} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>standard deviation <span class="s2">\( = 4.66\) <span class="Apple-converted-space"> </span></span><strong><em>A1</em></strong></p>
<p class="p3"> </p>
<p class="p2"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>A0 </em></strong>for \(4.77\).</p>
<p class="p2"><em><strong>[3 marks]</strong></em></p>
<p class="p2"><em><strong>Total [6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weights, in kg, of male birds of a certain species are modelled by a normal distribution with mean \(\mu \) and standard deviation \(\sigma \) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that 70 % of the birds weigh more than 2.1 kg and 25 % of the birds weigh more than 2.5 kg, calculate the value of \(\mu \) and the value of \(\sigma \) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A random sample of ten of these birds is obtained. Let <em>X</em> denote the number of birds in the sample weighing more than 2.5 kg.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Calculate \({\text{E}}(X)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Calculate the probability that exactly five of these birds weigh more than 2.5 kg.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Determine the most likely value of <em>X</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The number of eggs, <em>Y</em> , laid by female birds of this species during the nesting season is modelled by a Poisson distribution with mean \(\lambda \) . You are given that \({\text{P}}(Y \geqslant 2) = 0.80085\) , correct to 5 decimal places.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine the value of \(\lambda \) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Calculate the probability that two randomly chosen birds lay a total of</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">two eggs between them.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Given that the two birds lay a total of two eggs between them, calculate the probability that they each lay one egg.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">we are given that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2.1 = \mu - 0.5244\sigma \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2.5 = \mu + 0.6745\sigma \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu = 2.27{\text{ , }}\sigma = 0.334\) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) let <em>X</em> denote the number of birds weighing more than 2.5 kg</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">then <em>X</em> is B(10, 0.25) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(X) = 2.5\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) 0.0584 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) to find the most likely value of <em>X</em> , consider</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({p_0} = 0.0563 \ldots ,{\text{ }}{p_1} = 0.1877 \ldots ,{\text{ }}{p_2} = 0.2815 \ldots ,{\text{ }}{p_3} = 0.2502 \ldots \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">therefore, most likely value = 2 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) we solve \(1 - {\text{P}}(Y \leqslant 1) = 0.80085\) using the GDC <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\lambda = 3.00\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) let \({X_1}{\text{, }}{X_2}\) denote the number of eggs laid by each bird</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}({X_1} + {X_2} = 2) = {\text{P}}({X_1} = 0){\text{P}}({X_2} = 1) + {\text{P}}({X_1} = 1){\text{P}}({X_2} = 1) + {\text{P}}({X_1} = 2){\text{P}}({X_2} = 0)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {{\text{e}}^{ - 3}} \times {{\text{e}}^{ - 3}} \times \frac{9}{2} + {({{\text{e}}^{ - 3}} \times 3)^2} + {{\text{e}}^{ - 3}} \times \frac{9}{2} \times {{\text{e}}^{ - 3}} = 0.0446\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({\text{P}}({X_1} = 1,{\text{ }}{X_2} = 1|{X_1} + {X_2} = 2) = \frac{{{\text{P}}({X_1} = 1,{\text{ }}{X_2} = 1)}}{{{\text{P}}({X_1} + {X_2} = 2)}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.5\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A fisherman notices that in any hour of fishing, he is equally likely to catch exactly two fish, as he is to catch less than two fish. Assuming the number of fish caught can be modelled by a Poisson distribution, calculate the expected value of the number of fish caught when he spends four hours fishing.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{Po}}(m)\)</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 2) = {\text{P}}(X < 2)\) <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{1}{2}{m^2}{{\text{e}}^{ - m}} = {{\text{e}}^{ - m}}(1 + m)\) <strong><em>(A1)(A1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(m = 2.73 {\text{ }}\left( {1 + \sqrt 3 } \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">in four hours the expected value is 10.9\(\,\,\,\,\left( {4 + 4\sqrt 3 } \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Value of </span><em style="font-family: 'times new roman', times; font-size: medium;">m </em><span style="font-family: 'times new roman', times; font-size: medium;">does not need to be rounded.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span><strong style="font-size: 11px; line-height: normal;"><em><span style="font-family: 'times new roman', times; font-size: medium;">[5 marks]</span></em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates did not attempt this question and many others did not go beyond setting the equation up. Among the ones who attempted to solve the equation, once again, very few candidates took real advantage of GDC use to obtain the correct answer.</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A student arrives at a school \(X\) minutes after 08:00, where X may be assumed to be normally distributed. On a particular day it is observed that 40% of the students arrive before 08:30 and 90% arrive before 08:55.</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(f(x) = \frac{{\ln x}}{x}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , \(0 < x < {{\text{e}}^2}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the mean and standard deviation of \(X\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The school has 1200 students and classes start at 09:00. Estimate the number of students who will be late on that day.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Maelis had not arrived by 08:30. Find the probability that she arrived late.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">At 15:00 it is the end of the school day and it is assumed that the departure of the students from school can be modelled by a Poisson distribution. On average 24 students leave the school every minute.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the probability that at least 700 students leave school before 15:30.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">At 15:00 it is the end of the school day and it is assumed that the departure of the students from school can be modelled by a Poisson distribution. On average 24 students leave the school every minute.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There are 200 days in a school year. Given that \(Y\) denotes the number of days in the year that at least 700 students leave before 15:30, find<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \({\text{E}}(Y)\) ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(P(Y > 150)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(X < 30) = 0.4\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(X < 55) = 0.9\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">or relevant sketch <em><strong> (M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">given \(Z = \frac{{X - \mu }}{\sigma }\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(Z < z) = 0.4 \Rightarrow \frac{{30 - \mu }}{\sigma } = - 0.253...\) <em><strong> (A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(Z < z) = 0.9 \Rightarrow \frac{{55 - \mu }}{\sigma } = 1.28...\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\mu = 30 + \left( {0.253...} \right) \times \sigma = 55 - \left( {1.28...} \right) \times \sigma \) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\sigma = 16.3\) , \(\mu = 34.1\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Accept 16 and 34.</span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">N</span></strong><span style="font-family: times new roman,times; font-size: medium;"><strong>ote:</strong> Working with 830 and 855 will only gain the two <em><strong>M</strong></em> marks.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(X{\text{ ~ N}}\)(\(34.12…\), \(16.28...{^2}\) )</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">late to school \( \Rightarrow X > 60\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(X > 60) = 0.056...\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">number of students late \({\text{ = 0}}{\text{.0560}}... \times {\text{1200}}\) <em><strong>(M1)</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\(= 67\) (to nearest integer) A1</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Accept \(62\) for use of \(34\) and \(16\).</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(X > 60|X > 30) = \frac{{{\text{P}}(X > 60)}}{{{\text{P}}(X > 30)}}\) <em><strong>M1</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = 0.0935\)</span><span style="font-family: times new roman,times; font-size: medium;"> (accept anything between \(0.093\) and \(0.094\)) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> If \(34\) and \(16\) are used \(0.0870\) is obtained. This should be accepted.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">let \(L\) be the random variable of the number of students who leave school in a </span><span style="font-family: times new roman,times; font-size: medium;">30 minute interval</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">since \(24 \times 30 = 720\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{L ~ Po(720)}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(}}L \geqslant 700) = 1 - {\text{P}}(L \leqslant 699)\) <em><strong>(M1)</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = 0.777\)</span><span style="font-family: times new roman,times; font-size: medium;"> <strong><em>A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for \(P(L > 700) = 1 - P(L \leqslant 700)\) (this leads to \(0.765\)).</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(Y{\text{ ~ B}}\)(\(200\), \(0.7767…\)) <em><strong> (M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{E}}(Y) = 200 \times 0.7767... = 155\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>Note:</strong></em> On ft, use of \(0.765\) will lead to \(153\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({\text{P}}(Y > 150) = 1 - {\text{P}}(Y \leqslant 150)\) <em><strong>(M1)</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = 0.797\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Accept \(0.799\) from using rounded answer.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> On ft, use of \(0.765\) will lead to \(0.666\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates who had been prepared to solve questions from this part of the syllabus did well on the question. As a general point, candidates did not always write down clearly which distribution was being used. There were many candidates who seemed unfamiliar with the concept of Normal distributions as well as the Poisson and Binomial distributions and did not attempt the question. Parts (a) – (c) of the question were a variation on similar problems seen on previous examinations and there were a disappointing number of candidates who seemed unable to start the question. The use of 830 and 850 rather than minutes after 8am was seen and this caused students to lose marks despite knowing the method required. In general technology was used well and this was seen in (d) when solving a problem that involved a Poisson distribution. A number of candidates were unable to identify the Binomial distribution in (e).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates who had been prepared to solve questions from this part of the syllabus did well on the question. As a general point, candidates did not always write down clearly which distribution was being used. There were many candidates who seemed unfamiliar with the concept of Normal distributions as well as the Poisson and Binomial distributions and did not attempt the question. Parts (a) – (c) of the question were a variation on similar problems seen on previous examinations and there were a disappointing number of candidates who seemed unable to start the question. The use of 830 and 850 rather than minutes after 8am was seen and this caused students to lose marks despite knowing the method required. In general technology was used well and this was seen in (d) when solving a problem that involved a Poisson distribution. A number of candidates were unable to identify the Binomial distribution in (e).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates who had been prepared to solve questions from this part of the syllabus did well on the question. As a general point, candidates did not always write down clearly which distribution was being used. There were many candidates who seemed unfamiliar with the concept of Normal distributions as well as the Poisson and Binomial distributions and did not attempt the question. Parts (a) – (c) of the question were a variation on similar problems seen on previous examinations and there were a disappointing number of candidates who seemed unable to start the question. The use of 830 and 850 rather than minutes after 8am was seen and this caused students to lose marks despite knowing the method required. In general technology was used well and this was seen in (d) when solving a problem that involved a Poisson distribution. A number of candidates were unable to identify the Binomial distribution in (e).</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates who had been prepared to solve questions from this part of the syllabus did well on the question. As a general point, candidates did not always write down clearly which distribution was being used. There were many candidates who seemed unfamiliar with the concept of Normal distributions as well as the Poisson and Binomial distributions and did not attempt the question. Parts (a) – (c) of the question were a variation on similar problems seen on previous examinations and there were a disappointing number of candidates who seemed unable to start the question. The use of 830 and 850 rather than minutes after 8am was seen and this caused students to lose marks despite knowing the method required. In general technology was used well and this was seen in (d) when solving a problem that involved a Poisson distribution. A number of candidates were unable to identify the Binomial distribution in (e).</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates who had been prepared to solve questions from this part of the syllabus did well on the question. As a general point, candidates did not always write down clearly which distribution was being used. There were many candidates who seemed unfamiliar with the concept of Normal distributions as well as the Poisson and Binomial distributions and did not attempt the question. Parts (a) – (c) of the question were a variation on similar problems seen on previous examinations and there were a disappointing number of candidates who seemed unable to start the question. The use of 830 and 850 rather than minutes after 8am was seen and this caused students to lose marks despite knowing the method required. In general technology was used well and this was seen in (d) when solving a problem that involved a Poisson distribution. A number of candidates were unable to identify the Binomial distribution in (e).</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A mosaic is going to be created by randomly selecting <span class="s1">1000 </span>small tiles, each of which is either black or white. The probability that a tile is white is <span class="s1">0.1</span>. Let the random variable \(W\)<span class="s1"> </span>be the number of white tiles.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the distribution of \(W\), including the values of any parameters.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the mean of \(W\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\text{P}}(W > 89)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(W \sim B(1000,{\text{ }}0.1)\;\;\;\left( {{\text{accept }}C_k^{1000}{{(0.1)}^k}{{(0.9)}^{1000 - k}}} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1A1</em></strong></span></p>
<p class="p2"> </p>
<p class="p3"><strong>Note:</strong> <span class="Apple-converted-space"> </span>First <strong><em>A1</em></strong> is for recognizing the binomial, second <strong><em>A1</em></strong> for both parameters if stated explicitly in this part of the question.</p>
<p class="p3"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(\mu ( = 1000 \times 0.1) = 100\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P}}(W > 89) = {\text{P}}(W \ge 90)\;\;\;\left( { = 1 - {\text{P}}(W \le 89)} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\( = 0.867\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"> </p>
<p class="p3"><strong>Notes: <span class="Apple-converted-space"> </span></strong>Award <strong><em>M0A0</em></strong> for \(0.889\)</p>
<p class="p3"><em><strong>[2 marks]</strong></em></p>
<p class="p3"><em><strong>Total [5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Overall this question was well answered. In part (a) a number of candidates did not mention the binomial distribution or failed to state its parameters although they could go on and do the next parts.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b) most candidates could state the expected value.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (c) many candidates had problems with inequalities due to the discrete nature of the variable. Most candidates that could deal with the inequality were able to use the GDC to obtain the answer.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Ava and Barry play a game with a bag containing one green marble and two red marbles. Each player in turn randomly selects a marble from the bag, notes its colour and replaces it. Ava wins the game if she selects a green marble. Barry wins the game if he selects a red marble. Ava starts the game.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that Ava wins on her first turn.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that Barry wins on his first turn.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that Ava wins in one of her first three turns.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that Ava eventually wins.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P(Ava wins on her first turn)}} = \frac{1}{3}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P(Barry wins on his first turn)}} = {\left( {\frac{2}{3}} \right)^2}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\( = \frac{4}{9}\;\;\;( = 0.444)\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(P\)(Ava wins in one of her first three turns)</p>
<p class="p2">\( = \frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3}\) <span class="Apple-converted-space"> </span><strong><em>M1A1A1</em></strong></p>
<p class="p3"> </p>
<p class="p2"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>M1 </em></strong>for adding probabilities, award <strong><em>A1 </em></strong>for a correct second term and award <strong><em>A1 </em></strong>for a correct third term.</p>
<p class="p2">Accept a correctly labelled tree diagram, awarding marks as above.</p>
<p class="p3"> </p>
<p class="p2">\( = \frac{{103}}{{243}}\;\;\;( = 0.424)\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p2"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P(Ava eventually wins)}} = \frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3} + \left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\left( {\frac{2}{3}} \right)\left( {\frac{1}{3}} \right)\frac{1}{3} + \ldots \) <strong><em>(A1)</em></strong></p>
<p>using \({S_\infty } = \frac{a}{{1 - r}}\) with \(a = \frac{1}{3}\) and \(r = \frac{2}{9}\) <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for using \({S_\infty } = \frac{a}{{1 - r}}\) and award <strong><em>(A1) </em></strong>for \(a = \frac{1}{3}\) and \(r = \frac{2}{9}\).</p>
<p> </p>
<p>\( = \frac{3}{7}\;\;\;( = 0.429)\) <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<p><strong><em>Total [11 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a) and (b) were straightforward and were well done.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a) and (b) were straightforward and were well done.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (c) and (d) were also reasonably well done.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (c) and (d) were also reasonably well done. A pleasingly large number of candidates recognized that an infinite geometric series was required in part (d).</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: 'times new roman', times; font-size: medium;">The box and whisker plot below illustrates the IB grades obtained by 100 students.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-size: medium;"><img src="data:image/png;base64,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" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">IB grades can only take integer values.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">How many students obtained a grade of more than 4?</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">State, with reasons, the maximum possible number and minimum possible number of students who obtained a 4 in the exam.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">50 <strong><em>A1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[1 mark]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Lower quartile is 4 so at least 26 obtained a 4 <strong><em>R1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Lower bound is 26 <strong><em>A1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Minimum is 2 but the rest could be 4 <strong><em>R1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">So upper bound is 49 <strong><em>A1<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Do not allow follow through for </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A </em></strong><span style="font-family: 'times new roman', times; font-size: medium;">marks.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>If answers are incorrect award <strong><em>R0A0</em></strong>; if argument is correct but no clear lower/upper bound is stated award <strong><em>R1A0</em></strong><em>; </em>award <strong><em>R0A1 </em></strong>for correct answer without explanation or incorrect explanation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[4 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Very few candidates were successful in answering this question. In many cases it was clear that candidates were not familiar with box-and-whisker plots at all; in other cases the explanations given revealed various misconceptions.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Very few candidates were successful in answering this question. In many cases it was clear that candidates were not familiar with box-and-whisker plots at all; in other cases the explanations given revealed various misconceptions.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Six customers wait in a queue in a supermarket. A customer can choose to pay with cash or a credit card. Assume that whether or not a customer pays with a credit card is independent of any other customers’ methods of payment.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">It is known that 60% of customers choose to pay with a credit card.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the probability that:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) the first three customers pay with a credit card and the next three pay with cash;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) exactly three of the six customers pay with a credit card.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">There are <em>n </em>customers waiting in another queue in the same supermarket. The probability that at least one customer pays with cash is greater than 0.995.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the minimum value of <em>n</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) \({0.6^3} \times {0.4^3}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for use of the product of probabilities.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = 0.0138\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) binomial distribution \(X:{\text{B(6, 0.6)}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for recognizing the binomial distribution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \({\text{P}}(X = 3) = \) \(^6{C_3}{(0.6)^3}{(0.4)^3}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> \( = 0.276\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>(M1)A1 </em></strong>for \(^6{C_3} \times 0.0138 = 0.276\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(Y:{\text{B(}}n,{\text{ 0.4)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Y \geqslant 1) > 0.995\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(1 - {\text{P}}(Y = 0) > 0.995\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Y = 0) < 0.005\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for any of the last three lines. Accept equalities.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({0.6^n} < 0.005\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for attempting to solve \({0.6^n} < 0.005\) using any method, <em>eg</em>, logs, graphically, use of solver. Accept an equality.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(n > 10.4\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\therefore n = 11\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Jan and Sia have been selected to represent their country at an international discus throwing competition. Assume that the distance thrown by each athlete is normally distributed. The mean distance thrown by Jan in the past year was 60.33 metres with a standard deviation of 1.95 metres.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In the past year, 80 % of Jan’s throws have been longer than <em>x</em> metres. Find <em>x</em> correct to two decimal places.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In the past year, 80 % of Sia’s throws have been longer than 56.52 metres. If the mean distance of her throws was 59.39 metres, find the standard deviation of her throws.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This year, Sia’s throws have a mean of 59.50 metres and a standard deviation of 3.00 metres. The mean and standard deviation of Jan’s throws have remained the same. In the competition, an athlete must have at least one throw of 65 metres or more in the first round to qualify for the final round. Each athlete is allowed three throws in the first round.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine whether Jan or Sia is more likely to qualify for the final on their first throw.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the probability that both athletes qualify for the final.</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{N(60.33, 1.9}}{{\text{5}}^2})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X < x) = 0.2 \Rightarrow x = 58.69{\text{ m}}\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(z = - 0.8416 \ldots \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( - 0.8416 = \frac{{56.52 - 59.39}}{\sigma }\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sigma \approx 3.41\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Jan \(X \sim {\text{N(60.33, 1.9}}{{\text{5}}^2})\); Sia \(X \sim {\text{N(59.50, 3.0}}{{\text{0}}^2})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Jan: \({\text{P}}(X > 65) \approx 0.00831\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sia: \({\text{P}}(Y > 65) \approx 0.0334\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sia is more likely to qualify <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Only award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>R1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> if </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> has been awarded.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Jan: \({\text{P}}(X \geqslant 1) = 1 - {\text{P}}(X = 0)\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 1 - {(1 - 0.00831 \ldots )^3} \approx 0.0247\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sia: \({\text{P}}(Y \geqslant 1) = 1 - {\text{P}}(Y = 0) = 1 - {(1 - 0.0334 \ldots )^3} \approx 0.0968\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept 0.0240 and 0.0969.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">hence, \({\text{P}}(X \geqslant 1{\text{ and }}y \geqslant 1) = 0.0247 \times 0.0968 = 0.00239\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were generally accessible to many candidates. In (c)(i) quite a few candidates missed the wording ‘first throw’, and consequently in (ii) used the incorrect probabilities.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were generally accessible to many candidates. In (c)(i) quite a few candidates missed the wording ‘first throw’, and consequently in (ii) used the incorrect probabilities.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were generally accessible to many candidates. In (c)(i) quite a few candidates missed the wording ‘first throw’, and consequently in (ii) used the incorrect probabilities.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The times taken for male runners to complete a marathon can be modelled by a normal distribution with a mean 196 minutes and a standard deviation 24 minutes.</p>
</div>
<div class="specification">
<p>It is found that 5% of the male runners complete the marathon in less than \({T_1}\) minutes.</p>
</div>
<div class="specification">
<p>The times taken for female runners to complete the marathon can be modelled by a normal distribution with a mean 210 minutes. It is found that 58% of female runners complete the marathon between 185 and 235 minutes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that a runner selected at random will complete the marathon in less than 3 hours.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate \({T_1}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the standard deviation of the times taken by female runners.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(T \sim N(196,{\text{ }}{24^2})\)</p>
<p>\({\text{P}}(T < 180) = 0.252\) <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}(T < {T_1}) = 0.05\) <strong><em>(M1)</em></strong></p>
<p>\({T_1} = 157\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(F \sim N(210,{\text{ }}{\sigma ^2})\)</p>
<p>\({\text{P}}(F < 235) = 0.79\) <strong><em>(M1)</em></strong></p>
<p>\(\frac{{235 - 210}}{\sigma } = 0.806421\) or equivalent <strong><em>(M1)(A1)</em></strong></p>
<p>\(\sigma = 31.0\) <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Ahmed is typing Section A of a mathematics examination paper. The number of mistakes that he makes, <em>X</em> , can be modelled by a Poisson distribution with mean 3.2 . Find the probability that Ahmed makes exactly four mistakes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) His colleague, Levi, is typing Section B of this paper. The number of mistakes that he makes, <em>Y</em> , can be modelled by a Poisson distribution with mean <em>m</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) If \({\text{E}}({Y^2}) = 5.5\) , find the value of <em>m</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the probability that Levi makes exactly three mistakes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Given that <em>X</em> and <em>Y</em> are independent, find the probability that Ahmed makes exactly four mistakes and Levi makes exactly three mistakes.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(X \sim {\text{Po(3.2)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 4) = \frac{{{{\text{e}}^{ - 3.2}}{{3.2}^4}}}{{4!}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.178 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) \({\text{Var}}(Y) = {\text{E}}({Y^2}) - {{\text{E}}^2}(Y)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(m = 5.5 - {m^2}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>m</em> = 1.90 (or <em>m</em> = –2.90 which is valid) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(Y \sim {\text{Po(1.90)}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Y = 3) = \frac{{{{\text{e}}^{ - 1.90}}{{1.90}^4}}}{{3!}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.171 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Required probability \( = 0.171 \times 0.178 = 0.0304\) (accept 0.0305) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was correctly solved by most candidates, either using the formula or directly from their GDC. Solutions to (b), however, were extremely disappointing with the majority of candidates giving \(\sqrt 5 \), incorrectly, as their value of <em>m</em> . It was possible to apply follow through in (b) (ii) and (c) which were well done in general.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">A survey is conducted in a large office building. It is found that \(30\% \) of the office workers weigh less than \(62\) kg and that \(25\% \) of the office workers weigh more than \(98\) <span class="s1">kg.</span></p>
<p class="p2">The weights of the office workers may be modelled by a normal distribution with mean \(\mu \) and standard deviation \(\sigma \).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Determine two simultaneous linear equations satisfied by \(\mu \) and \(\sigma \).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the values of \(\mu \) and \(\sigma \).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that an office worker weighs more than <span class="s1">\(100\) </span>kg.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">There are elevators in the office building that take the office workers to their offices.</p>
<p class="p1">Given that there are <span class="s1">\(10\) </span>workers in a particular elevator,</p>
<p class="p1">find the probability that at least four of the workers weigh more than <span class="s1">\(100\) </span>kg.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that there are <span class="s1">\(10\) </span>workers in an elevator and at least one weighs more than <span class="s1">\(100\) </span>kg,</p>
<p class="p1">find the probability that there are fewer than four workers exceeding <span class="s1">\(100\) </span>kg.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The arrival of the elevators at the ground floor between <span class="s1">\(08:00\) </span>and <span class="s1">\(09:00\) </span>can be modelled by a Poisson distribution. Elevators arrive on average every <span class="s1">\(36\) </span>seconds.</p>
<p class="p1">Find the probability that in any half hour period between <span class="s1">\(08:00\) </span>and <span class="s1">\(09:00\) </span>more than <span class="s1">\(60\) </span>elevators arrive at the ground floor.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">An elevator can take a maximum of <span class="s1">\(10\) </span>workers. Given that <span class="s1">\(400\) </span>workers arrive in a half hour period independently of each other,</p>
<p class="p1">find the probability that there are sufficient elevators to take them to their offices.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>Note: </strong>In Section B, accept answers that correctly round to 2 sf.</p>
<p> </p>
<p>(i) let \(W\) be the weight of a worker and \(W \sim {\text{N}}(\mu ,{\text{ }}{\sigma ^2})\)</p>
<p>\({\text{P}}\left( {Z < \frac{{62 - \mu }}{\alpha }} \right) = 0.3\) and \({\text{P}}\left( {Z < \frac{{98 - \mu }}{\sigma }} \right) = 0.75\) <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>for a correctly shaded and labelled diagram.</p>
<p> </p>
<p>\(\frac{{62 - \mu }}{\sigma } = {\Phi ^{ - 1}}(0.3)\;\;\;( = - 0.524 \ldots )\;\;\;\)and</p>
<p>\(\frac{{98 - \mu }}{\sigma } = {\Phi ^{ - 1}}(0.75)\;\;\;( = 0.674 \ldots )\)</p>
<p>or linear equivalents <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Condone equations containing the GDC inverse normal command.</p>
<p> </p>
<p>(ii) attempting to solve simultaneously <strong><em>(M1)</em></strong></p>
<p>\(\mu = 77.7,{\text{ }}\sigma = 30.0\) <strong><em>A1A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>In Section B, accept answers that correctly round to 2 sf.</p>
<p class="p2"> </p>
<p class="p1">\({\text{P}}(W > 100) = 0.229\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>In Section B, accept answers that correctly round to 2 sf.</p>
<p class="p2"> </p>
<p class="p1">let \(X\) represent the number of workers over <span class="s1">\(100\) kg </span>in a lift of ten passengers</p>
<p class="p1">\(X \sim {\text{B}}(10,{\text{ }}0.229 \ldots )\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\({\text{P}}(X \ge 4) = 0.178\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>In Section B, accept answers that correctly round to 2 sf.</p>
<p class="p2"> </p>
<p class="p3">\({\text{P}}(X < 4|X \ge 1) = \frac{{{\text{P}}(1 \le X \le 3)}}{{{\text{P}}(X \ge 1)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1(A1)</em></strong></span></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award the <strong><em>M1 </em></strong>for a clear indication of a conditional probability.</p>
<p class="p2"> </p>
<p class="p3">\( = 0.808\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p3"><span class="s1"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>In Section B, accept answers that correctly round to 2 sf.</p>
<p class="p2"> </p>
<p class="p1">\(L \sim {\text{Po}}(50)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\({\text{P}}(L > 60) = 1 - {\text{P}}(L \le 60)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">\( = 0.0722\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>In Section B, accept answers that correctly round to 2 sf.</p>
<p class="p2"> </p>
<p class="p3"><span class="s1">\(400\)</span> workers require at least <span class="s1">\(40\)</span> <span class="s1">elevators <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></span></p>
<p class="p1">\({\text{P}}(L \ge 40) = 1 - {\text{P}}(L \le 39)\) <span class="Apple-converted-space"> </span><strong>(<em>M1)</em></strong></p>
<p class="p1">\( = 0.935\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [18 marks]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Farmer Suzie grows turnips and the weights of her turnips are normally distributed with a mean of \(122g\) and standard deviation of \(14.7g\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Calculate the percentage of Suzie’s turnips that weigh between \(110g\) and \(130g\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Suzie has <span class="s1">\(100\) </span>turnips to take to market. Find the expected number weighing more than \(130g\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Find the probability that at least \(30\) of the \(100g\) turnips weigh more than \(130g\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Farmer Ray also grows turnips and the weights of his turnips are normally distributed with a mean of \(144g\). Ray only takes to market turnips that weigh more than \(130g\). Over a period of time, Ray finds he has to reject \(1\) in \(15\) turnips due to their being underweight.</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the standard deviation of the weights of Ray’s turnips.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Ray has \(200\) turnips to take to market. Find the expected number weighing more than \(150g\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(i) \(P(110 < X < 130) = 0.49969 \ldots = 0.500 = 50.0\% \) <strong><em>(M1)A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept \(50\)</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1A0</em></strong> for \(0.50\) (\(0.500\))</p>
<p> </p>
<p>(ii) \(P(X > 130) = (1 - 0.707 \ldots ) = 0.293 \ldots \) <strong><em>M1</em></strong></p>
<p>expected number of turnips \( = 29.3\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept \(29\).</p>
<p> </p>
<p>(iii) no of turnips weighing more than \(130\) is \(Y \sim B(100,{\text{ }}0.293)\) <strong><em>M1</em></strong></p>
<p>\(P(Y \ge 30) = 0.478\) <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<p><strong><em> </em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(i) \(X \sim N(144,{\text{ }}{\sigma ^2})\)</p>
<p>\(P(X \le 130) = \frac{1}{{15}} = 0.0667\) <strong><em>(M1)</em></strong></p>
<p>\(P\left( {Z \le \frac{{130 - 144}}{\sigma }} \right) = 0.0667\)</p>
<p>\(\frac{{14}}{\sigma } = 1.501\) <strong><em>(A1)</em></strong></p>
<p>\(\sigma = 9.33{\text{ g}}\) <strong><em>A1</em></strong></p>
<p>(ii) \(P(X > 150|X > 130) = \frac{{P(X > 150)}}{{P(X > 130)}}\) <strong><em>M1</em></strong></p>
<p>\( = \frac{{0.26008 \ldots }}{{1 - 0.06667}} = 0.279\) <strong><em>A1</em></strong></p>
<p>expected number of turnips \( = 55.7\) <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<p><strong><em>Total [12 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Emma acquires a new cell phone for her birthday and receives texts from her friends. It is assumed that the daily number of texts Emma receives follows a Poisson distribution with mean \(m = 5\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the probability that on a certain day Emma receives more than \(7\) texts.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Determine the expected number of days in a week on which Emma receives more than \(7\) texts.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that Emma receives fewer than \(30\) texts during a week.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(X \sim Po(5)\)</p>
<p class="p1">\({\text{P}}(X \ge 8) = 0.133\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(7 \times 0.133 \ldots \) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p1">\( \approx 0.934{\text{ days}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"> </p>
<p class="p3"><strong>Note:</strong> <span class="Apple-converted-space"> </span>Accept “<span class="s2">\(1\) </span>day”.</p>
<p class="p3"><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(7 \times 5 = 35\;\;\;\left( {Y \sim Po(35)} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">\({\text{P}}(Y \le 29) = 0.177\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[3 marks]</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>Total [7 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The number of complaints per day received by customer service at a department store follows a Poisson distribution with a mean of \(0.6\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">On a randomly chosen day, find the probability that</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>there are no complaints;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>there are at least three complaints.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">In a randomly chosen five-day week, find the probability that there are no complaints.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">On a randomly chosen day, find the most likely number of complaints received.</p>
<p class="p1">Justify your answer.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The department store introduces a new policy to improve customer service. The number of complaints received per day now follows a Poisson distribution with mean \(\lambda \).</p>
<p class="p1">On a randomly chosen day, the probability that there are no complaints is now \(0.8\).</p>
<p class="p1">Find the value of \(\lambda \).</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(P(X = 0) = 0.549{\text{ }}( = {{\text{e}}^{ - 0.6}})\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(P(X \ge 3) = 1 - P(X \le 2)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\(P(X \ge 3) = 0.0231\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>EITHER</strong></p>
<p class="p1">using \(Y \sim {\text{Po(3)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">using \({(0.549)^5}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><strong>THEN</strong></p>
<p class="p1">\({\text{P}}(Y = 0) = 0.0498{\text{ }}\left( { = {{\text{e}}^{ - 3}}} \right)\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}(X = 0)\) (most likely number of complaints received is zero) <strong><em>A1</em></strong></p>
<p><strong>EITHER</strong></p>
<p>calculating \({\text{P}}(X = 0) = 0.549\) and \({\text{P}}(X = 1) = 0.329\) <strong><em>M1A1</em></strong></p>
<p><strong>OR</strong></p>
<p>sketching an appropriate (discrete) graph of \({\text{P}}(X = x)\) against \(x\) <strong><em>M1A1</em></strong></p>
<p><strong>OR</strong></p>
<p>finding \({\text{P}}(X = 0) = {e^{ - 0.6}}\) and stating that \({\text{P}}(X = 0) > 0.5\) <strong><em>M1A1</em></strong></p>
<p><strong>OR</strong></p>
<p>using \({\text{P}}(X = x) = {\text{P}}(X = x - 1) \times \frac{\mu }{x}\) where \(\mu < 1\) <strong><em>M1A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}(X = 0) = 0.8{\text{ (}} \Rightarrow {e^{ - \lambda }} = 0.8)\) <strong><em>(A1)</em></strong></p>
<p>\(\lambda = 0.223\left( { = \ln \frac{5}{4}, = - \ln \frac{4}{5}} \right)\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<p><strong><em>Total [10 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a), (b) and (d) were generally well done. In (a) (ii), some candidates calculated \(1 - {\text{P}}(X \le 3)\).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a), (b) and (d) were generally well done.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A number of candidates offered clear and well-reasoned solutions to part (c). The two most common successful approaches used to justify that the most likely number of complaints received is zero were either to calculate \({\text{P}}(X = x)\) for \(x = 0,{\text{ }}1,{\text{ }} \ldots \) or find that \({\text{P}}(X = 0) = 0.549{\text{ }}( > 0.5)\). A number of candidates stated that the most number of complaints received was the mean of the distribution \((\lambda = 0.6)\).</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Parts (a), (b) and (d) were generally well done.</p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The random variable \(X\) follows a Poisson distribution with mean \(m \ne 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(2{\text{P}}(X = 4) = {\text{P}}(X = 5)\), show that \(m = 10\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(X \le 11\), find the probability that \(X = 6\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(2\frac{{{{\text{e}}^{ - m}}{m^4}}}{{4!}} = \frac{{{{\text{e}}^{ - m}}{m^5}}}{{5!}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1A1</em></strong></span></p>
<p class="p1">\(\frac{2}{{4!}} = \frac{m}{{5!}}\;\;\;\)<span class="s1">or other simplification <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></span></p>
<p class="p2"> </p>
<p class="p3"><strong>Note:</strong> <span class="Apple-converted-space"> </span>accept a labelled graph showing clearly the solution to the equation. Do not accept simple verification that \(m = 10\) <span class="s2">is a solution.</span></p>
<p class="p4"> </p>
<p class="p1">\( \Rightarrow m = 10\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>AG</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P}}(X = 6|X \le 11) = \frac{{{\text{P}}(X = 6)}}{{{\text{P}}(X \le 11)}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1) (A1)</em></strong></span></p>
<p class="p1">\( = \frac{{0.063055 \ldots }}{{0.696776 \ldots }}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1">\( = 0.0905\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[4 marks]</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>Total [7 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates successfully finished part (a) with two fundamental errors occurring regularly. Either e was granted powers of 4 and 5 or an attempt to show that the value of \(m\) was 10 was made by evaluation.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b) was challenging for many candidates that showed that the idea of conditional probability was poorly understood. There were many incorrect solutions where often candidates only found \({\text{P}}(X = 6)\).</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>It is given that one in five cups of coffee contain more than 120 mg of caffeine.<br>It is also known that three in five cups contain more than 110 mg of caffeine.</p>
<p>Assume that the caffeine content of coffee is modelled by a normal distribution.<br>Find the mean and standard deviation of the caffeine content of coffee.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>let \(X\) be the random variable “amount of caffeine content in coffee”</p>
<p>\({\text{P}}(X > 120) = 0.2,{\text{ P}}(X > 110) = 0.6\) <strong><em>(M1)</em></strong></p>
<p>\(( \Rightarrow {\text{P}}(X < 120) = 0.8,{\text{ P}}(X < 110) = 0.4)\)</p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for at least one correct probability statement.</p>
<p> </p>
<p>\(\frac{{120 - \mu }}{\sigma } = 0.84162 \ldots ,{\text{ }}\frac{{110 - \mu }}{\sigma } = - 0.253347 \ldots \) <strong><em>(M1)(A1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for attempt to find at least one appropriate \(z\)-value.</p>
<p> </p>
<p>\(120 - \mu = 0.84162\sigma ,{\text{ }}110 - \mu = - 0.253347\sigma \)</p>
<p>attempt to solve simultaneous equations <strong><em>(M1)</em></strong></p>
<p>\(\mu = 112,{\text{ }}\sigma = 9.13\) <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A team of 6 players is to be selected from 10 volleyball players, of whom 8 are boys and 2 are girls.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In how many ways can the team be selected?</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In how many of these selections is exactly one girl in the team?</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If the selection of the team is made at random, find the probability that exactly </span><span style="font-family: 'times new roman', times; font-size: medium;">one girl is in the team.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(\left( {\begin{array}{*{20}{c}}<br> {10} \\ <br> 6<br>\end{array}} \right) = 210\) </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>(M1)A1</strong></em></span><br><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[2 marks]</strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(2 \times \left( {\begin{array}{*{20}{c}}<br> 8 \\ <br> 5 <br>\end{array}} \right) = 112\)</span><span style="font-family: times new roman,times; font-size: medium;"> <strong><em>(M1)A1A1</em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span><strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Accept \(210 - 28 - 70 = 112\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{112}}{{210}}\,\,\left( { = \frac{8}{{15}} = 0.533} \right)\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates answered this question well although in some cases candidates were not able to distinguish the use of permutations from combinations. Almost all candidates scored the two marks of part (c), but many of these were follow through marks.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates answered this question well although in some cases candidates were not able to distinguish the use of permutations from combinations. Almost all candidates scored the two marks of part (c), but many of these were follow through marks.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates answered this question well although in some cases candidates were not able to distinguish the use of permutations from combinations. Almost all candidates scored the two marks of part (c), but many of these were follow through marks.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A random variable \(X\) has a probability distribution given in the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-28_om_17.11.47.png" alt="N16/5/MATHL/HP2/ENG/TZ0/01"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine the value of \({\text{E}}({X^2})\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \({\text{Var}}(X)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({\text{E}}({X^2}) = \Sigma {x^2} \bullet {\text{P}}(X = x) = 10.37{\text{ }}( = 10.4{\text{ 3 sf)}}\) Â Â </span><strong><em>(M1)A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1"><span class="Apple-converted-space">\({\text{sd}}(X) = 1.44069 \ldots \) Â Â </span><strong><em>(M1)(A1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\({\text{Var}}(X) = 2.08{\text{ }}( = 2.0756)\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1"><span class="Apple-converted-space">\({\text{E}}(X) = 2.88{\text{ }}( = 0.06 + 0.27 + 0.5 + 0.98 + 0.63 + 0.44)\) Â Â </span><strong><em>(A1)</em></strong></p>
<p class="p1">use of \({\text{Var}}(X) = {\text{E}}({X^2}) - {\left( {{\text{E}}(X)} \right)^2}\)Â <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>(M1) </em></strong>only if \({\left( {{\text{E}}(X)} \right)^2}\) is used correctly.</p>
<p class="p2">Â </p>
<p class="p1">\(\left( {{\text{Var}}(X) = 10.37 - 8.29} \right)\)</p>
<p class="p1"><span class="Apple-converted-space">\({\text{Var}}(X) = 2.08{\text{ }}( = 2.0756)\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong>Accept <span class="s1">2.11</span>.</p>
<p class="p2">Â </p>
<p class="p1"><strong>METHOD 3</strong></p>
<p class="p1"><span class="Apple-converted-space">\({\text{E}}(X) = 2.88{\text{ }}( = 0.06 + 0.27 + 0.5 + 0.98 + 0.63 + 0.44)\) Â Â </span><strong><em>(A1)</em></strong></p>
<p class="p1">use of \({\text{Var}}(X) = {\text{E}}\left( {{{\left( {X - {\text{E}}(X)} \right)}^2}} \right)\)Â <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p1">\((0.679728 +  \ldots  + 0.549152)\)</p>
<p class="p1"><span class="Apple-converted-space">\({\text{Var}}(E) = 2.08{\text{ }}( = 2.0756)\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The weights, in kg, of one-year-old bear cubs are modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Given that the upper quartile weight is 21.3 kg and the lower quartile weight is 17.1 kg, calculate the value of \(\mu \) and the value of \(\sigma \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A random sample of 100 of these bear cubs is selected.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the expected number of bear cubs weighing more than 22 kg.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) <strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu = \frac{1}{2} \times (17.1 + 21.3)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu = 19.2{\text{ (kg)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">finding <em>z </em>value for the upper quartile \( = 0.674489{\text{K}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0.674489{\text{K}} = \frac{{21.3 - 19.2}}{\sigma }\) or \( - 0.674489{\text{K}} = \frac{{17.1 - 19.2}}{\sigma }\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sigma = 3.11{\text{ (kg)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">finding <em>z </em>value for the upper quartile \( = 0.674489{\text{K}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">from symmetry the <em>z </em>value for a lower quartile is \( - 0.674489{\text{K}}\) <em><strong>M1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">forming two simultaneous equations:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( - 0.674489{\text{K}} = \frac{{17.1 - \mu }}{\sigma }\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0.674489{\text{K}} = \frac{{21.3 - \mu }}{\sigma }\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solving gives:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu = 19.2{\text{ (kg)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sigma = 3.11{\text{ (kg)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) using \(100 \times {\text{P}}(X > 22) = 100 \times 0.184241{\text{K}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 18\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept 18.4</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [5 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A company produces computer microchips, which have a life expectancy that follows a normal distribution with a mean of 90 months and a standard deviation of 3.7 months.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) If a microchip is guaranteed for 84 months find the probability that it will fail before the guarantee ends.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The probability that a microchip does not fail before the end of the guarantee is required to be 99 %. For how many months should it be guaranteed?</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) A rival company produces microchips where the probability that they will fail after 84 months is 0.88. Given that the life expectancy also follows a normal distribution with standard deviation 3.7 months, find the mean.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \({\text{P}}(X \leqslant 84) = {\text{P}}(Z \leqslant - 1.62...) = 0.0524\) <strong><em>(M1)A1</em></strong> <strong><em>N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept 0.0526.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \({\text{P}}(Z \leqslant z) = 0.01 \Rightarrow z = - 2.326…\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \leqslant x) = {\text{P}}(Z \leqslant z) = 0.01 \Rightarrow z = - 2.326…\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 81.4\,\,\,\,\,{\text{(accept 81)}}\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \({\text{P}}(X \leqslant 84) = 0.12 \Rightarrow z = - 1.1749…\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{mean is 88.3}}\,\,\,\,\,{\text{(accept 88)}}\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A fair amount of students did not use their GDC directly, but used tables and more traditional methods to answer this question. Part (a) was answered correctly by most candidates using any method. A large number of candidates reversed the probabilities, i.e., failed to use a negative z value in parts (b) and (c), and hence did not obtain correct answers.</span></p>
</div>
<br><hr><br><div class="specification">
<p>There are 75 players in a golf club who take part in a golf tournament. The scores obtained on the 18th hole are as shown in the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_16.43.55.png" alt="M17/5/MATHL/HP2/ENG/TZ2/01"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>One of the players is chosen at random. Find the probability that this player’s score was 5 or more.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the mean score.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}(5{\text{ or more}}) = \frac{{29}}{{75}}\,\,\,( = 0.387)\) <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{mean score}} = \frac{{2 \times 3 + 3 \times 15 + 4 \times 28 + 5 \times 17 + 6 \times 9 + 7 \times 3}}{{75}}\) <strong><em>(M1)</em></strong></p>
<p>\( = \frac{{323}}{{75}}\,\,\,( = 4.31)\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The annual weather-related loss of an insurance company is modelled by a random </span><span style="font-family: times new roman,times; font-size: medium;">variable \(X\) with probability density function</span><span style="font-family: times new roman,times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {\frac{{2.5{{\left( {200} \right)}^{2.5}}}}{{{x^{3.5}}}},}&{x \geqslant 200} \\ <br> {0,}&{{\text{otherwise}}{\text{.}}} <br>\end{array}} \right.\]Find the median.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">\(\int_{200}^M {\frac{{2.5{{\left( {200} \right)}^{2.5}}}}{{{x^{3.5}}}}{\text{d}}x} = 0.5\) <em><strong>M1A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Award M1 for the integral equal to \(0.5\)</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"><em><strong> A1A1</strong></em> for the correct limits.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{ - {{200}^{2.5}}}}{{{M^{2.5}}}}\left( {\frac{{ - {{200}^{2.5}}}}{{{{200}^{2.5}}}}} \right) = 0.5\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1A1A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Award <em><strong>M1</strong></em> for correct integration</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>A1A1</strong></em> for correct substitutions.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{ - {{200}^{2.5}}}}{{{M^{2.5}}}} + 1 = 0.5 \Rightarrow {M^{2.5}} = 2{\left( {200} \right)^{2.5}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(M = 264\) <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[8 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Many students used incorrect limits to the integral, although many did correctly let the integral equal to \(0.5\).</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">Natasha lives in Chicago and has relatives in Nashville and St. Louis.</p>
<p class="p1">Each time she visits her relatives, she either flies or drives.</p>
<p class="p1">When travelling to Nashville, the probability that she drives is \(\frac{4}{5}\), and when travelling to St. Louis, the probability that she flies is \(\frac{1}{3}\)<span class="s1">.</span></p>
<p class="p2">Given that the probability that she drives when visiting her relatives is <span class="s2">\(\frac{13}{18}\)</span>, find the probability that for a particular trip,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">she travels to Nashville;</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">she is on her way to Nashville, given that she is flying.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><img src="images/Schermafbeelding_2016-01-06_om_17.06.16.png" alt></p>
<p class="p2">attempt to set up the problem using a tree diagram and/or an equation, with the unknown \(x\) <strong><em>M1</em></strong></p>
<p class="p1">\(\frac{4}{5}x + \frac{2}{3}(1 - x) = \frac{{13}}{{18}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1">\(\frac{{4x}}{5} - \frac{{2x}}{3} = \frac{{13}}{{18}} - \frac{2}{3}\)</p>
<p class="p1">\(\frac{{2x}}{{15}} = \frac{1}{{18}}\)</p>
<p class="p1">\(x = \frac{5}{{12}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><img src="images/Schermafbeelding_2016-01-06_om_17.06.16_1.png" alt></p>
<p class="p2">attempt to set up the problem using conditional probability <span class="Apple-converted-space"> </span><strong><em>M1</em></strong></p>
<p class="p2"><strong>EITHER</strong></p>
<p class="p1">\(\frac{{\frac{5}{{12}} \times \frac{1}{5}}}{{1 - \frac{{13}}{{18}}}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong>OR</strong></p>
<p class="p1">\(\frac{{\frac{5}{{12}} \times \frac{1}{5}}}{{\frac{1}{{12}} + \frac{7}{{36}}}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong>THEN</strong></p>
<p class="p1">\( = \frac{3}{{10}}\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[3 marks]</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>Total [6 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Events \(A\) and \(B\) are such that \({\text{P}}(A \cup B) = 0.95,{\text{ P}}(A \cap B) = 0.6\) and \({\text{P}}(A|B) = 0.75\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{P}}(B)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{P}}(A)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that events \(A’\) and \(B\) are independent.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}(A|B) = \frac{{{\text{P}}(A \cap B)}}{{{\text{P}}(B)}}\)</p>
<p>\( \Rightarrow 0.75 = \frac{{0.6}}{{{\text{P}}(B)}}\) <strong><em>(M1)</em></strong></p>
<p>\( \Rightarrow {\text{P}}(B){\text{ }}\left( { = \frac{{0.6}}{{0.75}}} \right) = 0.8\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A \cap B)\)</p>
<p>\( \Rightarrow 0.95 = {\text{P}}(A) + 0.8 - 0.6\) <strong><em>(M1)</em></strong></p>
<p>\( \Rightarrow {\text{P}}(A) = 0.75\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>\({\text{P}}(A'|B) = \frac{{{\text{P}}(A' \cap B)}}{{{\text{P}}(B)}} = \frac{{0.2}}{{0.8}} = 0.25\) <strong><em>A1</em></strong></p>
<p>\({\text{P}}(A'|B) = {\text{P}}(A’)\) <strong><em>R1</em></strong></p>
<p>hence \(A’\) and \(B\) are independent <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>Note:</strong> If there is evidence that the student has calculated \({\text{P}}(A' \cap B) = 0.2\) by assuming independence in the first place, award <strong><em>A0R0</em></strong>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><strong>EITHER</strong></p>
<p>\({\text{P}}(A) = {\text{P}}(A|B)\) <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p>\({\text{P}}(A) \times {\text{P}}(B) = 0.75 \times 0.80 = 0.6 = {\text{P}}(A \cap B)\) <strong><em>A1</em></strong></p>
<p><strong>THEN</strong></p>
<p>\(A\) and \(B\) are independent <strong><em>R1</em></strong></p>
<p>hence \(A’\) and \(B\) are independent <strong><em>AG</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p>\({\text{P}}(A') \times {\text{P}}(B) = 0.25 \times 0.80 = 0.2\) <strong><em>A1</em></strong></p>
<p>\({\text{P}}(A') \times {\text{P}}(B) = {\text{P}}(A' \cap B)\) <strong><em>R1</em></strong></p>
<p>hence \(A’\) and \(B\) are independent <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> follows a Poisson distribution with mean \(\lambda \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find \(\lambda \) if \({\text{P}}(X = 0) + {\text{P}}(X = 1) = 0.123\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) With this value of \(\lambda \), find \({\text{P}}(0 < X < 9)\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) required to solve \({{\text{e}}^{ - \lambda }} + \lambda {{\text{e}}^{ - \lambda }} = 0.123\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">solving to obtain \(\lambda = 3.63\) <strong><em>A2</em></strong> <strong><em>N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A2</em></strong> if an additional negative solution is seen but <strong><em>A0</em></strong> if only a negative solution is seen.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \({\text{P}}(0 < X < 9)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\text{P}}(X \leqslant 8) - {\text{P}}(X = 0)\) (or equivalent) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.961\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) - Well done by most, although there were some answers that ignored the requirement of mathematical notation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (b) - Not successfully answered by many. The main problem was not correctly interpreting the inequalities in the probability.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A continuous random variable <em>X</em> has a probability density function given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {\frac{{{{(x + 1)}^3}}}{{60}},}&{{\text{for }}1 \leqslant x \leqslant 3} \\ <br> {0,}&{{\text{otherwise}}{\text{.}}} <br>\end{array},} \right.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \({\text{P}}(1.5 \leqslant X \leqslant 2.5)\) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) E(<em>X</em>) ;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) the median of <em>X</em> .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\int_{1.5}^{2.5} {\frac{{{{(x + 1)}^3}}}{{60}}{\text{d}}x = 0.4625\,\,\,\,\,{\text{( = 0.463)}}} \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \({\text{E}}(X) = \int_1^3 {\frac{{x{{(x + 1)}^3}}}{{60}}{\text{d}}x = 2.31} \) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \(\int_1^m {\frac{{{{(x + 1)}^3}}}{{60}}{\text{d}}x = 0.5} \) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left[ {\frac{{{{(x + 1)}^4}}}{{240}}} \right]_1^m = 0.5\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(m = 2.41\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were reasonably well done in general but (c) caused problems for many candidates where several misconceptions regarding the median were seen. The expectation was that candidates would use their GDCs to solve (a) and (b), and possibly even (c), although in the event most candidates did the integrations by hand. Those candidates using their GDCs made fewer mistakes in general than those doing the integrations analytically.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X </em>has the distribution \({\text{Po}}(m)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \({\text{P}}(X = 5) = {\text{P}}(X = 3) + {\text{P}}(X = 4)\), find</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">the value of <em>m </em>;</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">P (<em>X </em>> 2) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 5) = {\text{P}}(X = 3) + {\text{P}}(X = 4)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{{\text{e}}^{ - m}}{m^5}}}{{5!}} = \frac{{{{\text{e}}^{ - m}}{m^3}}}{{3!}} + \frac{{{{\text{e}}^{ - m}}{m^4}}}{{4!}}\) <strong> <em>M1(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({m^2} - 5m - 20 = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow m = \frac{{5 + \sqrt {105} }}{2} = (7.62)\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[3 marks]</span><br></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X > 2) = 1 - {\text{P}}(X \leqslant 2)\) <strong> <em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 1 - 0.018...\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.982\) <strong> <em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[2 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Again this proved to be a successful question for many candidates with a good proportion of wholly correct answers seen. It was good to see students making good use of the calculator.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Again this proved to be a successful question for many candidates with a good proportion of wholly correct answers seen. It was good to see students making good use of the calculator.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The fish in a lake have weights that are normally distributed with a mean of 1.3 kg and a standard deviation of 0.2 kg.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the probability that a fish which is caught weighs less than 1.4 kg.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">John catches 6 fish. Calculate the probability that at least 4 of the fish weigh more than 1.4 kg.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine the probability that a fish which is caught weighs less than 1 kg, given that it weighs less than 1.4 kg.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(x < 1.4) = 0.691\,\,\,\,\,\)(accept 0.692) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(y \sim {\text{B(6, 0.3085…)}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(Y \geqslant 4) = 1 - {\text{P}}(Y \leqslant 3)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.0775\,\,\,\,\,\)(accept 0.0778 if 3sf approximation from (a) used) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{B(6, 0.6914…)}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \leqslant 2)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.0775\,\,\,\,\,\)(accept 0.0778 if 3sf approximation from (a) used) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(x < 1|x < 1.4) = \frac{{{\text{P}}(x < 1)}}{{{\text{P}}(x < 1.4)}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{0.06680…}}{{0.6914…}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.0966\,\,\,\,\,\)(accept 0.0967) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was almost universally correctly answered, albeit with an accuracy penalty in some cases. In (b) it was generally recognised that the distribution was binomial, but with some wavering about the correct value of the parameter <em>p</em>. Part (c) was sometimes answered correctly, but not with much confidence.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was almost universally correctly answered, albeit with an accuracy penalty in some cases. In (b) it was generally recognised that the distribution was binomial, but with some wavering about the correct value of the parameter <em>p</em>. Part (c) was sometimes answered correctly, but not with much confidence.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was almost universally correctly answered, albeit with an accuracy penalty in some cases. In (b) it was generally recognised that the distribution was binomial, but with some wavering about the correct value of the parameter <em>p</em>. Part (c) was sometimes answered correctly, but not with much confidence.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A company produces rectangular sheets of glass of area <span class="s1">5 </span>square metres. During manufacturing these glass sheets flaws occur at the rate of <span class="s1">0.5 </span>per <span class="s1">5 </span>square metres. It is assumed that the number of flaws per glass sheet follows a Poisson distribution.</p>
</div>
<div class="specification">
<p class="p1">Glass sheets with no flaws earn a profit of <span class="s1">$5</span>. Glass sheets with at least one flaw incur a loss of <span class="s1">$3</span>.</p>
</div>
<div class="specification">
<p class="p1">This company also produces larger glass sheets of area <span class="s1">20 </span>square metres. The rate of occurrence of flaws remains at <span class="s1">0.5 </span>per <span class="s1">5 </span>square metres.</p>
<p class="p1">A larger glass sheet is chosen at random.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that a randomly chosen glass sheet contains at least one flaw.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the expected profit, \(P\) dollars, per glass sheet.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the probability that it contains no flaws.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({\text{X}} \sim {\text{Po}}(0.5)\) Â Â </span><strong><em>(A1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{P}}(X \geqslant 1) = 0.393{\text{ }}( = 1 - {{\text{e}}^{ - 0.5}})\) Â Â </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(X = 0) = 0.607 \ldots \) Â Â </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\({\text{E}}(P) = (0.607 \ldots  \times 5) - (0.393 \ldots  \times 3)\)   </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">the expected profit is $1.85 <span class="s1">per glass sheet <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(Y \sim {\text{Po}}(2)\) Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{P}}(Y = 0) = 0.135{\text{ }}( = {{\text{e}}^{ - 2}})\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was reasonably well done. Some candidates calculated \({\text{P}}(X = 1)\).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b) was not as well done as expected with a surprising number of candidates calculating \(5{\text{P}}(X = 0) + 3{\text{P}}(X \geqslant 1)\) rather than \(5{\text{P}}(X = 0) - 3{\text{P}}(X \geqslant 1)\).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (c) was very well done.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Each of the 25 students in a class are asked how many pets they own. Two students own three pets and no students own more than three pets. The mean and standard deviation of the number of pets owned by students in the class are \(\frac{{18}}{{25}}\) and \(\frac{{24}}{{25}}\) respectively.</p>
<p>Find the number of students in the class who do not own a pet.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p>let <em>p</em> have no pets, <em>q</em> have one pet and <em>r</em> have two pets   <em><strong> (M1)</strong></em></p>
<p><em>p</em>Â + <em>q</em>Â + <em>r</em>Â + 2Â = 25Â Â Â <em><strong>(A1)</strong></em></p>
<p>0<em>p</em>Â + 1<em>q</em>Â + 2<em>r</em>Â + 6Â = 18Â Â Â <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept a statement that there are a total of 12 pets.</p>
<p>attempt to use variance equation, or evidence of trial and error    <em><strong>(M1)</strong></em></p>
<p>\(\frac{{0p + 1q + 4r + 18}}{{25}} - {\left( {\frac{{18}}{{25}}} \right)^2} = {\left( {\frac{{24}}{{25}}} \right)^2}\)Â Â Â <em><strong>(A1)</strong></em></p>
<p>attempt to solve a system of linear equations <em><strong>(M1)</strong></em></p>
<p><em>p</em>Â = 14Â Â Â <em><strong>A1</strong></em></p>
<p>Â </p>
<p><strong>METHOD 2</strong></p>
<p><img src="data:image/png;base64,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">Â Â Â <em><strong>(M1)</strong></em></p>
<p>\(p + q + r + \frac{2}{{25}} = 1\)Â Â Â <em><strong>Â (A1)</strong></em></p>
<p>\(q + 2r + \frac{6}{{25}} = \frac{{18}}{{25}}\left( { \Rightarrow q + 2r = \frac{{12}}{{25}}} \right)\)Â Â Â <em><strong>A1</strong></em></p>
<p>\(q + 4r + \frac{{18}}{{25}} - {\left( {\frac{{18}}{{25}}} \right)^2} = \frac{{576}}{{625}}\left( { \Rightarrow q + 4r = \frac{{18}}{{25}}} \right)\)Â Â Â Â <em><strong>(M1)(A1)</strong></em></p>
<p>\(q = \frac{6}{{25}},\,\,r = \frac{3}{{25}}\)Â Â Â Â <em><strong>(M1)</strong></em></p>
<p>\(p = \frac{{14}}{{25}}\)Â Â Â Â <em><strong>A1</strong></em></p>
<p>so 14 have no pets</p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p class="p1">The heights of students in a single year group in a large school can be modelled by a normal distribution.</p>
<p class="p1">It is given that 40% of the students are shorter than 1.62 m and 25% are taller than 1.79 m<span class="s1">.</span></p>
<p class="p2">Find the mean and standard deviation of the heights of the students.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>let the heights of the students be \(X\)</p>
<p>\({\text{P}}(X < 1.62) = 0.4,{\text{ P}}(X > 1.79) = 0.25\) Â Â <strong><em>M1</em></strong></p>
<p><strong>Note: Â Â </strong>Award <strong><em>M1 </em></strong>for either of the probabilities above.</p>
<p>\({\text{P}}\left( {Z < \frac{{1.62 - \mu }}{\sigma }} \right) = 0.4,{\text{ P}}\left( {Z < \frac{{1.79 - \mu }}{\sigma }} \right) = 0.75\) Â Â <strong><em>M1</em></strong></p>
<p><strong>Note: Â Â </strong>Award <strong><em>M1 </em></strong>for either of the expressions above.</p>
<p>\(\frac{{1.62 - \mu }}{\sigma } =Â - 0.2533 \ldots ,{\text{ }}\frac{{1.79 - \mu }}{\sigma } = 0.6744 \ldots \) Â Â <strong><em>M1A1</em></strong></p>
<p><strong>Note: Â Â <em>A1 </em></strong>for both values correct.</p>
<p>\(\mu = 1.67{\text{ (m)}},{\text{ }}\sigma = 0.183{\text{ (m)}}\)   <strong><em>A1A1</em></strong></p>
<p><strong>Note: Â Â </strong>Accept answers that round to 1.7 (m) and 0.18 (m).</p>
<p><strong>Note: </strong>Accept answers in centimetres.</p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">A large number of good solutions in this question, although candidates failing on the question failed at different stages. A number did not standardise the distribution correctly, and there were others who were unable to correctly solve the simultaneous equations. There were a notable number of otherwise good candidates who were unable to attempt the question, even though it is of a very standard type.</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The duration of direct flights from London to Singapore in a particular year followed a normal distribution with mean \(\mu \) and standard deviation \(\sigma \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">92% of flights took under 13 hours, while only 12% of flights took under 12 hours 35 minutes.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\mu \) and \(\sigma \) to the nearest minute.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}\left( {Z < \frac{{780 - \mu }}{\sigma }} \right) = 0.92\) and \({\text{P}}\left( {Z < \frac{{755 - \mu }}{\sigma }} \right) = 0.12\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">use of inverse normal <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \frac{{780 - \mu }}{\sigma } = 1.405 \ldots \) and \(\frac{{755 - \mu }}{\sigma } = - 1.174 \ldots \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">solving simultaneously <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1 </em></strong>for attempting to solve an incorrect pair of equations <em>eg, </em>inverse normal not used.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu = 766.385\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sigma = 9.6897\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu = 12{\text{ hrs 46 mins (}} = 766{\text{ mins)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sigma = 10{\text{ mins}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Generally well done. Most candidates made correct use of the symmetry of the normal curve and the inverse normal to set up a correct pair of equations involving \(\mu \) and \(\sigma \). A few candidates expressed equations containing the GDC command term invNorm.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A few candidates did not express their answers correct to the nearest minute and a few candidates performed erroneous conversions from hours to minutes.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Emily walks to school every day. The length of time this takes can be modelled by a normal distribution with a mean of 11 minutes and a standard deviation of 3 minutes. She is late if her journey takes more than 15 minutes.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability she is late next Monday.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability she is late at least once during the next week (Monday to Friday).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>X</em> represent the length of time a journey takes on a particular day.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(P(X > 15) = 0.0912112819 \ldots = 0.0912\) <strong><em>(M1)A1</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use of correct Binomial distribution <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(N \sim B(5,0.091 \ldots )\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(1 - 0.0912112819 \ldots = 0.9087887181 \ldots \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(1 - {(0.9087887181 \ldots )^5} = 0.380109935 \ldots = 0.380\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Allow answers to be given as percentages.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were many good answers to this question. Some students lost accuracy marks by early rounding. Some students struggled with the Binomial distribution.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were many good answers to this question. Some students lost accuracy marks by early rounding. Some students struggled with the Binomial distribution.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Over a one month period, Ava and Sven play a total of <em>n</em> games of tennis.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The probability that Ava wins any game is 0.4. The result of each game played is independent of any other game played.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>X</em> denote the number of games won by Ava over a one month period.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find an expression for P(<em>X</em> = 2) in terms of <em>n</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) If the probability that Ava wins two games is 0.121 correct to three decimal places, find the value of <em>n</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(X \sim {\text{B}}(n,{\text{ }}0.4)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using \({\text{P}}(X = x) = \left( {\begin{array}{*{20}{c}}<br> n \\ <br> r <br>\end{array}} \right){(0.4)^x}{(0.6)^{n - x}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 2) = \left( {\begin{array}{*{20}{c}}<br> n \\ <br> 2 <br>\end{array}} \right){(0.4)^2}{(0.6)^{n - 2}}\) \(\left( { = \frac{{n(n - 1)}}{2}{{(0.4)}^2}{{(0.6)}^{n - 2}}} \right)\) <strong><em>A1</em></strong> <strong><em>N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) P(<em>X</em> = 2) = 0.121 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using an appropriate method (including trial and error) to solve their equation. <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>n</em> = 10 <strong><em>A1</em></strong> <strong><em>N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Do not award the last <strong><em>A1</em></strong> if any other solution is given in their final answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was generally well done. The most common error was to omit the binomial coefficient <em>i.e.</em> not identifying that the situation is described by a binomial distribution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Finding the correct value of <em>n</em> in part (b) proved to be more elusive. A significant proportion of candidates attempted algebraic approaches and seemingly did not realise that the equation could only be solved numerically. Candidates who obtained <em>n</em> = 10 often accomplished this by firstly attempting to solve the equation algebraically before ‘resorting’ to a GDC approach. Some candidates did not specify their final answer as an integer while others stated <em>n</em> = 1.76 as their final answer.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The number of birds seen on a power line on any day can be modelled by a Poisson distribution with mean 5.84.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that during a certain seven-day week, more than 40 birds have been seen on the power line.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">On Monday there were more than 10 birds seen on the power line. Show that the probability of there being more than 40 birds seen on the power line from that Monday to the following Sunday, inclusive, can be expressed as:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{P}}(X > 40) + \sum\limits_{r = 11}^{40} {{\text{P}}(X = r){\text{P}}(Y > 40 - r)} }}{{{\text{P}}(X > 10)}}\) where \(X \sim {\text{Po}}(5.84)\) and \(Y \sim {\text{Po}}(35.04)\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">mean for week is 40.88 <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(S > 40) = 1 - {\text{P}}(S \leqslant 40) = 0.513\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{probability there were more than 10 on Monday AND more than 40 over the week}}}}{{{\text{probability there were more than 10 on Monday}}}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">possibilities for the numerator are:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">there were more than 40 birds on the power line on Monday <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">11 on Monday and more than 29 over the course of the next 6 days <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">12 on Monday and more than 28 over the course of the next 6 days … until</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">40 on Monday and more than 0 over the course of the next 6 days <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">hence if <em>X </em>is the number on the power line on Monday and <em>Y</em>, the number on the power line Tuesday – Sunday then the numerator is <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X > 40) + {\text{P}}(X = 11) \times {\text{P}}(Y > 29) + {\text{P}}(X = 12) \times {\text{P}}(Y > 28) + \ldots \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( + {\text{P}}(X = 40) \times {\text{P}}(Y > 0)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\text{P}}(X > 40) + \sum\limits_{r = 11}^{40} {{\text{P}}(X = r){\text{P}}(Y > 40 - r)} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">hence solution is \(\frac{{{\text{P}}(X > 40) + \sum\limits_{r = 11}^{40} {{\text{P}}(X = r){\text{P}}(Y > 40 - r)} }}{{{\text{P}}(X > 10)}}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Mr Lee is planning to go fishing this weekend. Assuming that the number of fish caught per hour follows a Poisson distribution with mean \(0.6\), find</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) the probability that he catches at least one fish in the first hour;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) the probability that he catches exactly three fish if he fishes for four hours;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) the number of complete hours that Mr Lee needs to fish so that the probability </span><span style="font-family: times new roman,times; font-size: medium;">of catching more than two fish exceeds 80 %.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) \(X{\text{ ~ Po(0}}{\text{.6)}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(X \geqslant 1) = 1 - {\text{P}}(X = 0)\) <em><strong>M1</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = 0.451\) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) \(Y{\text{ \~ Po(2}}{\text{.4)}}\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(Y = 3) = 0.209\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) \(Z{\text{ \~ Po(}}0.6n{\text{)}}\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(Z \geqslant 3) = 1 - {\text{P}}(Z \leqslant 2) > 0.8\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Only one of these <em><strong>M1</strong></em> marks may be implied.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(n \geqslant 7.132...\) (hours)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so, Mr Lee needs to fish for at least \(8\) complete hours <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Accept a shown trial and error method that leads to a correct solution.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">It was clear that many students had not been taught the topic and were consequently unable to make an attempt at the question. Of those students who were able to start, common errors were in a misunderstanding of the language. Many had difficulties in part (c) and “at least” in part (a) was sometimes misinterpreted.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) A box of biscuits is considered to be underweight if it weighs less than 228 grams. It is known that the weights of these boxes of biscuits are normally distributed with a mean of 231 grams and a standard deviation of 1.5 grams.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">What is the probability that a box is underweight?</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) The manufacturer decides that the probability of a box being underweight should be reduced to 0.002.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Bill’s suggestion is to increase the mean and leave the standard deviation unchanged. Find the value of the new mean.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Sarah’s suggestion is to reduce the standard deviation and leave the mean unchanged. Find the value of the new standard deviation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) After the probability of a box being underweight has been reduced to 0.002, a group of customers buys 100 boxes of biscuits. Find the probability that at least two of the boxes are underweight.</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">Part A.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">There are six boys and five girls in a school tennis club. A team of two boys and two girls will be selected to represent the school in a tennis competition.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) In how many different ways can the team be selected?</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Tim is the youngest boy in the club and Anna is the youngest girl. In how many different ways can the team be selected if it must include both of them?</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) What is the probability that the team includes both Tim and Anna?</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Fred is the oldest boy in the club. Given that Fred is selected for the team, what is the probability that the team includes Tim or Anna, but not both?</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">Part B.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(X \sim {\text{N(231, 1.}}{{\text{5}}^2})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X < 228) = 0.0228\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept 0.0227.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) \(X \sim {\text{N(}}\mu {\text{, 1.}}{{\text{5}}^2})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X < 228) = 0.002\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{228 - \mu }}{{1.5}} = - 2.878…\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu = 232{\text{ grams}}\) <strong><em>A1</em></strong> <strong><em>N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(X \sim {\text{N(231, }}{\sigma ^2})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{228 - 231}}{\sigma } = - 2.878…\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sigma = 1.04{\text{ grams}}\) <strong><em>A1</em></strong> <strong><em>N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \(X \sim {\text{B(100, 0.002)}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \leqslant 1) = 0.982…\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \geqslant 2) = 1 - {\text{P}}(X \leqslant 1) = 0.0174\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [11 marks]</em></strong></span></p>
<div class="question_part_label">Part A.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Boys can be chosen in \(\frac{{6 \times 5}}{2} = 15\) ways <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Girls can be chosen in \(\frac{{5 \times 4}}{2} = 10\) ways <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Total \( = 15 \times 10 = 150\) ways <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Number of ways \( = 5 \times 4 = 20\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) \(\frac{{20}}{{150}}{\text{ }}\left( { = \frac{2}{{15}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) <strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(T) = \frac{1}{5};{\text{ P}}(A) = \frac{2}{5}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">P(<em>T</em> or <em>A</em> but not both) \( = {\text{P}}(T) \times {\text{P}}(A') + {\text{P}}(T') \times {\text{P}}(A)\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{5} \times \frac{3}{5} + \frac{4}{5} \times \frac{2}{5} = \frac{{11}}{{25}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Number of selections including Fred \( = 5 \times \left( {\begin{array}{*{20}{c}}<br> 5 \\ <br> 2 <br>\end{array}} \right) = 50\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Number of selections including Tim but not Anna \( = \left( {\begin{array}{*{20}{c}}<br> 4 \\ <br> 2 <br>\end{array}} \right) = 6\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Number of selections including Anna but not Tim \( = 4 \times 4 = 16\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Both statements are needed to award <strong><em>A1</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">P(<em>T</em> or <em>A</em> but not both) \( = \frac{{6 + 16}}{{50}} = \frac{{11}}{{25}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [10 marks]</em></strong></span></p>
<div class="question_part_label">Part B.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Part A was well done by many candidates although an arithmetic penalty was often awarded in (b)(i) for giving the new value of the mean to too many significant figures.</span></p>
<div class="question_part_label">Part A.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Candidates are known, however, to be generally uncomfortable with combinatorial mathematics and Part B caused problems for many candidates. Even some of those candidates who solved (a) and (b) correctly were then unable to deduce the answer to (c), sometimes going off on some long-winded solution which invariably gave the wrong answer. Very few correct solutions were seen to (d).</span></p>
<div class="question_part_label">Part B.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider two events \(A\) and \(B\) such that \({\text{P}}(A) = k,{\text{ P}}(B) = 3k,{\text{ P}}(A \cap B) = {k^2}\) and \({\text{P}}(A \cup B) = 0.5\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate \(k\);</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \({\text{P}}(A' \cap B)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>use of \({\text{P}}(A \cup B) = {\text{P}}(A) + {\text{P}}(B) - {\text{P}}(A \cap B)\) <strong><em>M1</em></strong></p>
<p>\(0.5 = k + 3k - {k^2}\) <strong><em>A1</em></strong></p>
<p>\({k^2} - 4k + 0.5 = 0\)</p>
<p>\(k = 0.129\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award the final <strong><em>A1 </em></strong>if two solutions are given.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of \({\text{P}}(A' \cap B) = {\text{P}}(B) - {\text{P}}(A \cap B)\) or alternative <strong><em>(M1)</em></strong></p>
<p>\({\text{P}}(A' \cap B) = 3k - {k^2}\) <strong><em>(A1)</em></strong></p>
<p>\( = 0.371\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>When carpet is manufactured, small faults occur at random. The number of faults in Premium carpets can be modelled by a Poisson distribution with mean 0.5 faults per 20\(\,\)m<sup>2</sup>. Mr Jones chooses Premium carpets to replace the carpets in his office building. The office building has 10 rooms, each with the area of 80\(\,\)m<sup>2</sup>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that the carpet laid in the first room has fewer than three faults.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that exactly seven rooms will have fewer than three faults in the carpet.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(\lambda = 4 \times 0.5\) <strong><em>(M1)</em></strong></p>
<p>\(\lambda = 2\) <strong><em>(A1)</em></strong></p>
<p>\({\text{P}}(X \leqslant 2) = 0.677\) <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(Y \sim B(10,{\text{ }}0,677)\) <strong><em>(M1)(A1)</em></strong></p>
<p>\({\text{P}}(Y = 7) = 0.263\) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for clear recognition of binomial distribution.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The probability density function of the continuous random variable <em>X </em>is given by <br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {k{2^{\frac{1}{x}}},}&{1 \leqslant x \leqslant 2} \\ <br> {0,}&{{\text{otherwise}}} <br>\end{array}} \right.\] <br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">where <em>k </em>is a constant. Find the expected value of <em>X </em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: 'times new roman', times; font-size: medium;">\(k\int_1^2 {{2^{\frac{1}{x}}}{\text{d}}x = 1 \Rightarrow k = \frac{1}{{\int_1^2 {{2^{\frac{1}{x}}}{\text{d}}x} }}{\text{ }}( = 0.61556...)} \) <strong><em>(M1)(A1)</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}(X) = k\int_1^2 {x{2^{\frac{1}{x}}}{\text{d}}x = 2.39....k{\text{ or 1.47}}} \) <strong><em>M1A1</em></strong></span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"> </span><strong style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">Condone missing d</span><em style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">x </em><span style="font-family: 'times new roman', times; font-size: medium; line-height: normal;">in any part of the question.</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Times;"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[4 marks]</span><br></em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was well attempted by most candidates. However many were not alert for the necessity of using GDC to calculate the definite integrals and wasted time trying to obtain these values using standard calculus methods without success.</span></p>
</div>
<br><hr><br><div class="question">
<p>The mean number of squirrels in a certain area is known to be 3.2 squirrels per hectare of woodland. Within this area, there is a 56 hectare woodland nature reserve. It is known that there are currently at least 168 squirrels in this reserve.</p>
<p>Assuming the population of squirrels follow a Poisson distribution, calculate the probability that there are more than 190 squirrels in the reserve.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><em>X</em> is number of squirrels in reserve<br><em>X</em> ∼ Po(179.2)   <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> A1</strong></em> if 179.2 or 56 × 3.2 seen or implicit in future calculations.</p>
<p>recognising conditional probability   <em><strong>M1</strong></em></p>
<p>P(<em>X</em> > 190 | <em>X</em> ≥ 168)</p>
<p>\( = \frac{{{\text{P}}\left( {X > 190} \right)}}{{{\text{P}}\left( {X \geqslant 168} \right)}} = \left( {\frac{{0.19827 \ldots }}{{0.80817 \ldots }}} \right)\)Â Â Â Â <em><strong>(A1)(A1)</strong></em></p>
<p>=Â 0.245Â Â Â <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">A set of 15 observations has mean 11.5 and variance 9.3. One observation of 22.1 is considered unreliable and is removed. Find the mean and variance of the remaining 14 observations.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{\sum\limits_{i = 1}^{15} {{x_i}} }}{{15}} = 11.5 \Rightarrow \sum\limits_{i = 1}^{15} {{x_i} = 172.5} \) <strong> <em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 11.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">new mean \(= \frac{{172.5 - 22.1}}{{14}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 10.7428… = 10.7 (3sf) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{\sum\limits_{i = 1}^{15} {x_i^2} }}{{15}} - {11.5^2} = 9.3\) <em><strong>(M1)</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \sum\limits_{i = 1}^{15} {x_i^2 = 2123.25} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">new variance \(= \frac{{2123.25 - {{22.1}^2}}}{{14}} - {(10.7428...)^2}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 1.37 (3sf) <em><strong>A1</strong></em><br></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates were successful in finding the correct value of the mean; however, the variance caused many difficulties. Many candidates affirmed that there were no differences in the variance as it remained constant; some others got wrong results due to premature rounding of figures. Many candidates lost the final mark because they rounded their answers prematurely, resulting in a very inaccurate answer to this question.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The probability density function of a random variable <em>X </em>is defined as:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: normal;">\[f(x) = \left\{ \begin{array}{r}ax\cos x,\\0,\end{array} \right.\begin{array}{*{20}{l}}{0 \le x \le {\textstyle{\pi \over 2}},{\rm{where }}\,a \in \mathbb{R}}\\{{\rm{elsewhere}}}\end{array}\]</span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that \(a = \frac{2}{{\pi - 2}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find \({\text{P}}\left( {X < \frac{\pi }{4}} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Find:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) the mode of <em>X</em>;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) the median of <em>X</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Find \({\text{P}}\left( {X < \frac{\pi }{8}|X < \frac{\pi }{4}} \right)\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(a\int_0^{\frac{\pi }{2}} {x\cos x{\text{d}}x = 1} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">integrating by parts:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(u = x\) \(v' = \cos x\)</span><span style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"> </span><strong style="font-family: 'times new roman', times; font-size: medium; background-color: #f7f7f7;"><em>M1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(u' = 1\) \(v = \sin x\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {x\cos x{\text{d}}x = x\sin x + \cos x} \) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left[ {x\sin x + \cos x} \right]_0^{\frac{\pi }{2}} = \frac{\pi }{2} - 1\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = \frac{1}{{\frac{\pi }{2} - 1}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{2}{{\pi - 2}}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \({\text{P}}\left( {X < \frac{\pi }{4}} \right) = \frac{2}{{\pi - 2}}\int_0^{\frac{\pi }{4}} {x\cos x{\text{d}}x = 0.460} \) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept \(\frac{2}{{\pi - 2}}{\text{ }}\left( { = \frac{{\pi \sqrt 2 }}{8} + \frac{{\sqrt 2 }}{2} - 1} \right)\) or equivalent</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) \({\text{mode}} = 0.860\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (<em>x</em>-value of a maximum on the graph over the given domain)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) \(\frac{2}{{\pi - 2}}\int_0^m {x\cos x{\text{d}}x = 0.5} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(\int_0^m {x\cos x{\text{d}}x = \frac{{\pi - 2}}{4}} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(m\sin m + \cos m - 1 = \frac{{\pi - 2}}{4}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \({\text{median}} = 0.826\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Do not accept answers containing additional solutions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) \({\text{P}}\left( {X < \frac{\pi }{8}|X < \frac{\pi }{4}} \right) = \frac{{{\text{P}}\left( {X < \frac{\pi }{8}} \right)}}{{{\text{P}}\left( {X < \frac{\pi }{4}} \right)}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{0.129912}}{{0.459826}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.283\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [13 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The random variable<em> X</em> has a binomial distribution with parameters <em>n</em> and <em>p</em>.<br>It is given that E(<em>X</em>) = 3.5.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the least possible value of <em>n</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>It is further given that P(<em>X</em> ≤ 1) = 0.09478 correct to 4 significant figures.</p>
<p>Determine the value of <em>n</em> and the value of <em>p</em>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em>np</em>Â = 3.5Â Â Â <em><strong>(A1)</strong></em></p>
<p><em>p </em>≤ 1 ⇒ least <em>n</em> = 4    <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(1 − <em>p</em>)<em><sup>n</sup></em> + <em>np</em>(1 − <em>p</em>)<sup><em>n</em>−1</sup> = 0.09478   <em><strong>M1A1</strong></em></p>
<p>attempt to solve above equation with <em>np</em>Â = 3.5Â Â Â <em><strong>(M1)</strong></em></p>
<p><em>n</em> = 12, <em>p</em> = \(\frac{7}{{24}}\) (=0.292)   <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Do not accept <em>n</em> as a decimal.</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A discrete random variable \(X\) follows a Poisson distribution \({\text{Po}}(\mu )\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \({\text{P}}(X = x + 1) = \frac{\mu }{{x + 1}} \times {\text{P}}(X = x),{\text{ }}x \in \mathbb{N}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Given that \({\text{P}}(X = 2) = 0.241667\)Â </span>and \({\text{P}}(X = 3) = 0.112777\), use part (a) to find the value of \(\mu \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(X = x + 1) = \frac{{{\mu ^{x + 1}}}}{{(x + 1)!}}{{\text{e}}^{ - \mu }}\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{\mu }{{x + 1}} \times \frac{{{\mu ^x}}}{{x!}}{{\text{e}}^{ - \mu }}\) Â Â </span><strong><em>M1A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{\mu }{{x + 1}} \times {\text{P}}(X = x)\) Â Â </span><strong><em>AG</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1"><span class="Apple-converted-space">\(\frac{\mu }{{x + 1}} \times {\text{P}}(X = x) = \frac{\mu }{{x + 1}} \times \frac{{{\mu ^x}}}{{x!}}{{\text{e}}^{ - \mu }}\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{{\mu ^{x + 1}}}}{{(x + 1)!}}{{\text{e}}^{ - \mu }}\) Â Â </span><strong><em>M1A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = {\text{P}}(X = x + 1)\) Â Â </span><strong><em>AG</em></strong></p>
<p class="p1"><strong>METHOD 3</strong></p>
<p class="p1"><span class="Apple-converted-space">\(\frac{{{\text{P}}(X = x + 1)}}{{{\text{P}}(X = x)}} = \frac{{\frac{{{\mu ^{x + 1}}}}{{(x + 1)!}}{{\text{e}}^{ - \mu }}}}{{\frac{{{\mu ^x}}}{{x!}}{{\text{e}}^{ - \mu }}}}\) Â Â </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{{\mu ^{x + 1}}}}{{{\mu ^x}}} \times \frac{{x!}}{{(x + 1)!}}\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{\mu }{{x + 1}}\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1">and so \({\text{P}}(X = x + 1) = \frac{\mu }{{x + 1}} \times {\text{P}}(X = x)\)Â <span class="Apple-converted-space">Â Â </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(X = 3) = \frac{\mu }{3} \bullet {\text{P}}(X = 2){\text{ }}\left( {0.112777 = \frac{\mu }{3} \bullet 0.241667} \right)\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1">attempting to solve for \(\mu \)Â <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(\mu  = 1.40\)   </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">A random variable \(X\) </span>is normally distributed with mean 3 and variance \({2^2}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\text{P}}(0 \leqslant X \leqslant 2)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \({\text{P}}(\left| X \right| > 1)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">If \({\text{P}}(X > c) = 0.44\)</span>, find the value of \(c\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(0 \leqslant X \leqslant 2) = 0.242\) Â Â </span><span class="s1"><strong><em>(M1)A1</em></strong></span></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{P}}(\left| X \right| > 1) = {\text{P}}(X < Â - 1) + {\text{P}}(X > 1)\) Â Â </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\( = 0.02275 \ldots  + 0.84134 \ldots \)   </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\( = 0.864\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p2"><span class="Apple-converted-space">\({\text{P}}(\left| X \right| > 1) = 1 - {\text{P}}( - 1 < X < 1)\) Â Â </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\( = 1 - 0.13590 \ldots \) Â Â </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><span class="Apple-converted-space">\( = 0.864\) Â Â </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\(c = 3.30\) Â Â </span><strong><em>(M1)A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was generally well done. In each question part, a number of candidates could have benefited from producing a labelled sketch of the situation.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (b) was not well done with many candidates not knowing what \({\text{P}}(\left| X \right| > 1)\) represents. In each question part, a number of candidates could have benefited from producing a labelled sketch of the situation.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (c), a number of candidates did not recognise that \({\text{P}}(X > c) = 1 - {\text{P}}(X \leqslant c)\). In each question part, a number of candidates could have benefited from producing a labelled sketch of the situation.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> follows a Poisson distribution with mean <em>m</em> and satisfies</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{\text{P}}(X = 1) + {\text{P}}(X = 3) = {\text{P}}(X = 0) + {\text{P}}(X = 2).\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the value of <em>m</em> correct to four decimal places.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) For this value of <em>m</em>, calculate \({\text{P}}(1 \leqslant X \leqslant 2)\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \({\text{P}}(X = 1) + {\text{P}}(X = 3) = {\text{P}}(X = 0) + {\text{P}}(X = 2)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(m{{\text{e}}^{ - m}} + \frac{{{m^3}{{\text{e}}^{ - m}}}}{6} = {{\text{e}}^{ - m}} + \frac{{{m^2}{{\text{e}}^{ - m}}}}{2}\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({m^3} - 3{m^2} + 6m - 6 = 0\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(m = 1.5961\) (4 decimal places) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(m = 1.5961 \Rightarrow {\text{P}}(1 \leqslant X \leqslant 2) = m{{\text{e}}^{ - m}} + \frac{{{m^2}{{\text{e}}^{ - m}}}}{2} = 0.582\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates correctly stated the required equation for <em>m</em>. However, many algebraic errors in the simplification of this equation led to incorrect answers. Also, many candidates failed to find the value of <em>m</em> to the required accuracy, with many candidates giving answers correct to 4 sf instead of 4 dp. In part (b) many candidates did not realize that they needed to calculate \({\text{P}}(X = 1) + {\text{P}}(X = 2)\) and many attempts to calculate other combinations of probabilities were seen.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A student sits a national test and is told that the marks follow a normal distribution with mean 100. The student receives a mark of 124 and is told that he is at the \({68^{{\text{th}}}}\) percentile.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the variance of the distribution.</span></p>
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<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X:{\text{N}}(100,{\text{ }}{\sigma ^2})\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X < 124) = 0.68\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{24}}{\sigma } = 0.4676 \ldots \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\sigma = 51.315 \ldots \) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">variance = 2630 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes:</strong> Accept use of \({\text{P}}(X < 124.5) = 0.68\) leading to variance = 2744.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
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<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">It is believed that the lifespans of Manx cats are normally distributed with a mean of 13.5 years and a variance of 9.5 \({\text{year}}{{\text{s}}^2}\).</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the range of lifespans of Manx cats whose lifespans are within one standard deviation of the mean.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Estimate the number of Manx cats in a population of 10 000 that will have a lifespan of less than 10 years. Give your answer to the nearest whole number.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim N(13.5,{\text{ }}9.5)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(13.5 - \sqrt {9.5} < X < 13.5 + \sqrt {9.5} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(10.4 < X < 16.6\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept 6.16.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X < 10) = 0.12807 \ldots \) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">estimate is 1281 (correct to the nearest whole number). <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Accept 1280.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
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<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A large proportion of candidates experienced difficulties with this question. In parts (a) and (b), the most common error was to use <em>σ </em>= 9.5. In part (a), a large number of candidates used their range of values to then unnecessarily find the corresponding probability of that time interval occurring. In part (b), a large number of candidates used an unrealistic lower bound (a large negative value) for time.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A large proportion of candidates experienced difficulties with this question. In parts (a) and (b), the most common error was to use <em>σ </em>= 9.5. In part (a), a large number of candidates used their range of values to then unnecessarily find the corresponding probability of that time interval occurring. In part (b), a large number of candidates used an unrealistic lower bound (a large negative value) for time.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Bob measured the heights of 63 students. After analysis, he conjectured that the height, \(H\) , of the students could be modelled by a normal distribution with mean 166.5 cm and standard deviation 5 cm.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Based on this assumption, estimate the number of these students whose height is at least 170 cm.<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Later Bob noticed that the tape he had used to measure the heights was faulty as it started at the 5 cm mark and not at the zero mark.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) What are the correct values of the mean and variance of the distribution of the heights of these students?</span></p>
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<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) \(H{\text{ \~ N}}\)(\(166.5\), \({5^2}\) )</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(H \geqslant 170) = 0.242...\) <em><strong>(M1)(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.242... \times 63 = 15.2\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">so, approximately \(15\) students</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) correct mean: \(161.5\) (cm) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">variance remains the same, <em>i.e.</em> 25 (cm<sup>2</sup>) <em><strong>A2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">A surprising number of students lacked the basic knowledge of the normal distribution and </span><span style="font-family: times new roman,times; font-size: medium;">were unable to answer the first part of this question. Those students who showed a </span><span style="font-family: times new roman,times; font-size: medium;">knowledge of the topic tended to answer the question well. In part (b) many students either </span><span style="font-family: times new roman,times; font-size: medium;">had a misunderstanding of the difference between variance and standard deviation, or did not </span><span style="font-family: times new roman,times; font-size: medium;">read the question properly.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">At the start of each week, Eric and Marina pick a night at random on which they will watch a movie.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">If they choose a Saturday night, the probability that they watch a French movie is \(\frac{7}{9}\) and if they choose any other night the probability that they watch a French movie is \(\frac{4}{9}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that they watch a French movie.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that last week they watched a French movie, find the probability that it was on a Saturday night.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(F) = \left( {\frac{1}{7} \times \frac{7}{9}} \right) + \left( {\frac{6}{7} \times \frac{4}{9}} \right)\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1 </em></strong>for the sum of two products.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{31}}{{63}}{\text{ }}( = 0.4920 \ldots )\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Use of \({\text{P}}(S|F) = \frac{{{\text{P}}(S \cap F)}}{{{\text{P}}(F)}}\) to obtain \({\text{P}}(S|F) = \frac{{\frac{1}{7} \times \frac{7}{9}}}{{\frac{{31}}{{63}}}}\). <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1 </em></strong>only if the numerator results from the product of two probabilities.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{7}{{31}}{\text{ }}( = 0.2258 \ldots )\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Both parts were very well done. In part (a), most candidates successfully used a tree diagram.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Both parts were very well done. In part (b), most candidates correctly used conditional probability considerations.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The marks obtained by a group of students in a class test are shown below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given the mean of the marks is 6.5, find the value of <em>k</em>.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica; min-height: 34.0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{5 \times 6 + 6k + 7 \times 3 + 8 \times 1 + 9 \times 2 + 10 \times 1}}{{13 + k}} = 6.5\) (or equivalent) <strong><em>(M1)(A1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>(M1)(A1)</em></strong> for correct numerator, and <strong><em>(A1)</em></strong> for correct denominator.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(0.5k = 2.5 \Rightarrow k = 5\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The question was well done generally as one would expect.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">A Chocolate Shop advertises free gifts to customers that collect three vouchers. The vouchers are placed at random into 10% of all chocolate bars sold at this shop. Kati buys some of these bars and she opens them one at a time to see if they contain a voucher. Let \({\text{P}}(X = n)\) be the probability that Kati obtains her third voucher on the \(n{\text{th}}\) <span class="s1">bar opened.</span></p>
<p class="p1">(It is assumed that the probability that a chocolate bar contains a voucher stays at 10% throughout the question.)</p>
</div>
<div class="specification">
<p class="p1">It is given that \({\text{P}}(X = n) = \frac{{{n^2} + an + b}}{{2000}} \times {0.9^{n - 3}}\) for \(n \geqslant 3,{\text{ }}n \in \mathbb{N}\).</p>
</div>
<div class="specification">
<p class="p1">Kati’s mother goes to the shop and buys \(x\) chocolate bars. She takes the bars home for Kati to open.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{P}}(X = 3) = 0.001\) and \({\text{P}}(X = 4) = 0.0027\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the values of the constants \(a\) <span class="s1">and \(b\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce that \(\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{0.9(n - 1)}}{{n - 3}}\) for \(n > 3\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space">Â Â </span>Hence show that \(X\) has two modes \({m_1}\) and \({m_2}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space">Â Â </span>State the values of \({m_1}\) and \({m_2}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Determine the minimum value of \(x\) </span>such that the probability Kati receives at least one free gift is greater than <span class="s2">0.5.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(X = 3) = {(0.1)^3}\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = 0.001\) Â Â </span><strong><em>AG</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(X = 4) = {\text{P}}(VV\bar VV) + {\text{P}}(V\bar VVV) + {\text{P}}(\bar VVVV)\) Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p1">\( = 3 \times {(0.1)^3} \times 0.9\) (or equivalent) <span class="Apple-converted-space">Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = 0.0027\) Â Â </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">attempting to form equations in \(a\) and \(b\) <span class="Apple-converted-space">Â Â </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(\frac{{9 + 3a + b}}{{2000}} = \frac{1}{{1000}}{\text{ }}(3a + b = Â - 7)\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(\frac{{16 + 4a + b}}{{2000}} \times \frac{9}{{10}} = \frac{{27}}{{10\,000}}{\text{ }}(4a + b = Â - 10)\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1">attempting to solve simultaneously <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(a = Â - 3,{\text{ }}b = 2\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(X = n) = \left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) \times {0.1^3} \times {0.9^{n - 3}}\) Â Â </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{(n - 1)(n - 2)}}{{2000}} \times {0.9^{n - 3}}\) Â Â </span><strong><em>(M1)A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{{n^2} - 3n + 2}}{{2000}} \times {0.9^{n - 3}}\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(a = Â - 3,b = 2\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong>Condone the absence of \({0.9^{n - 3}}\) in the determination of the values of \(a\) and \(b\).</p>
<p class="p2">Â </p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1"><strong>EITHER</strong></p>
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(X = n) = \frac{{{n^2} - 3n + 2}}{{2000}} \times {0.9^{n - 3}}\) Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p1"><strong>OR</strong></p>
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(X = n) = \left( {\begin{array}{*{20}{c}} {n - 1} \\ 2 \end{array}} \right) \times {0.1^3} \times {0.9^{n - 3}}\) Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p1"><strong>THEN</strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{(n - 1)(n - 2)}}{{2000}} \times {0.9^{n - 3}}\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\({\text{P}}(X = n - 1) = \frac{{(n - 2)(n - 3)}}{{2000}} \times {0.9^{n - 4}}\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{(n - 1)(n - 2)}}{{(n - 2)(n - 3)}} \times 0.9\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{0.9(n - 1)}}{{n - 3}}\) Â Â </span><strong><em>AG</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1"><span class="Apple-converted-space">\(\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{\frac{{{n^2} - 3n + 2}}{{2000}} \times {{0.9}^{n - 3}}}}{{\frac{{{{(n - 1)}^2} - 3(n - 1) + 2}}{{2000}} \times {{0.9}^{n - 4}}}}\) Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{0.9({n^2} - 3n + 2)}}{{({n^2} - 5n + 6)}}\) Â Â </span><strong><em>A1A1</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>A1 </em></strong>for a correct numerator and <strong><em>A1 </em></strong>for a correct denominator.</p>
<p class="p2">Â </p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{0.9(n - 1)(n - 2)}}{{(n - 2)(n - 3)}}\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\( = \frac{{0.9(n - 1)}}{{n - 3}}\) Â Â </span><strong><em>AG</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space">Â Â </span>attempting to solve \(\frac{{0.9(n - 1)}}{{n - 3}} = 1\) for \(n\) <span class="Apple-converted-space">Â Â </span><strong><em>M1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(n = 21\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(\frac{{0.9(n - 1)}}{{n - 3}} < 1 \Rightarrow n > 21\) Â Â </span><strong><em>R1</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(\frac{{0.9(n - 1)}}{{n - 3}} > 1 \Rightarrow n < 21\) Â Â </span><strong><em>R1</em></strong></p>
<p class="p1">\(X\) has two modes <span class="Apple-converted-space">Â Â </span><strong><em>AG</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>R1R1 </em></strong>for a clearly labelled graphical representation of the two inequalities (using \(\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}}\)).</p>
<p class="p2">Â </p>
<p class="p1">(ii) <span class="Apple-converted-space">Â Â </span>the modes are <span class="s1">20 </span>and <span class="s1">21 <span class="Apple-converted-space">Â Â </span></span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1"><span class="Apple-converted-space">\(Y \sim {\text{B}}(x,{\text{ }}0.1)\) Â Â </span><strong><em>(A1)</em></strong></p>
<p class="p1">attempting to solve \({\text{P}}(Y \geqslant 3) > 0.5\) (or equivalent <em>eg</em>Â \(1 - {\text{P}}(Y \leqslant 2) > 0.5\)) for \(x\) <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p2">Â </p>
<p class="p1"><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for attempting to solve an equality (obtaining \(x = 26.4\)).</p>
<p class="p2">Â </p>
<p class="p1"><span class="Apple-converted-space">\(x = 27\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1"><span class="Apple-converted-space">\(\sum\limits_{n = 0}^x {{\text{P}}(X = n) > 0.5} \) Â Â </span><strong><em>(A1)</em></strong></p>
<p class="p1">attempting to solve for \(x\) <span class="Apple-converted-space">Â Â </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="Apple-converted-space">\(x = 27\) Â Â </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>The number of taxis arriving at Cardiff Central railway station can be modelled by a Poisson distribution. During busy periods of the day, taxis arrive at a mean rate of 5.3 taxis every 10 minutes. Let T represent a random 10 minute busy period.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that exactly 4 taxis arrive during T.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the most likely number of taxis that would arrive during T.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that more than 5 taxis arrive during T, find the probability that exactly 7 taxis arrive during T.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>During quiet periods of the day, taxis arrive at a mean rate of 1.3 taxis every 10 minutes.</p>
<p>Find the probability that during a period of 15 minutes, of which the first 10 minutes is busy and the next 5 minutes is quiet, that exactly 2 taxis arrive.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(X \sim {\text{Po}}\left( {5.3} \right)\)</p>
<p>\({\text{P}}\left( {X = 4} \right) = {{\text{e}}^{ - 5.3}}\frac{{{{5.3}^4}}}{{4{\text{!}}}}\)Â Â Â <em><strong>(M1)</strong></em></p>
<p>=Â 0.164Â Â Â <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>listing probabilities (table or graph)Â Â Â <em><strong>M1</strong></em></p>
<p>mode <em>X</em>Â = 5 (with probability 0.174)Â Â Â <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M0A0</strong></em> for 5 (taxis) or mode = 5 with no justification.</p>
<p>Â </p>
<p><strong>METHOD 2</strong></p>
<p>mode is the integer part of mean   <em><strong>R1</strong></em></p>
<p>E(<em>X</em>) = 5.3 ⇒ mode = 5   <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not allow <em><strong>R0A1</strong></em>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt at conditional probability    <em><strong>(M1)</strong></em></p>
<p>\(\frac{{{\text{P}}\left( {X = 7} \right)}}{{{\text{P}}\left( {X \geqslant 6} \right)}}\) or equivalent \(\left( { = \frac{{0.1163 \ldots }}{{0.4365 \ldots }}} \right)\)    <em><strong>A1</strong></em></p>
<p>= 0.267Â Â Â <em><strong>Â A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>METHOD 1</strong></em></p>
<p>the possible arrivals are (2,0), (1,1), (0,2)Â Â Â Â <em><strong>(A1)</strong></em></p>
<p>\(Y \sim {\text{Po}}\left( {0.65} \right)\)Â Â Â <em><strong>A1</strong></em></p>
<p>attempt to compute, using sum and product rule,   <em><strong>(M1)</strong></em></p>
<p>0.070106… × 0.52204… + 0.026455… × 0.33932… + 0.0049916… × 0.11028…    <em><strong>(A1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for one correct product and <em><strong>A1</strong> </em>for two other correct products.</p>
<p>= 0.0461Â Â Â Â <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<p>Â </p>
<p><strong>METHOD 2</strong></p>
<p>recognising a sum of 2 independent Poisson variables <em>eg Z</em>Â = <em>X</em> + <em>Y</em>Â Â Â <em><strong>R1</strong></em></p>
<p>\(\lambda = 5.3 + \frac{{1.3}}{2}\)</p>
<p>P(<em>Z</em> = 2) = 0.0461Â Â Â <em><strong>(M1)A3</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<p>Â </p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Express the sum of the first <em>n</em> positive odd integers using sigma notation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that the sum stated above is \({n^2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Deduce the value of the difference between the sum of the first 47 positive odd integers and the sum of the first 14 positive odd integers.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A number of distinct points are marked on the circumference of a circle, forming a polygon. Diagonals are drawn by joining all pairs of non-adjacent points.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show on a diagram all diagonals if there are 5 points.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that the number of diagonals is \(\frac{{n(n - 3)}}{2}\) if there are n points, where \(n > 2\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Given that there are more than one million diagonals, determine the least number of points for which this is possible.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable \(X \sim B(n,{\text{ }}p)\) has mean 4 and variance 3.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine <em>n</em> and <em>p</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the probability that in a single experiment the outcome is 1 or 3.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(\sum\limits_{k = 1}^n {(2k - 1)} \) (or equivalent) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A0</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for \(\sum\limits_{n = 1}^n {(2n - 1)} \) or equivalent.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) </span><strong style="font-family: 'times new roman', times; font-size: medium;">EITHER</strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2 \times \frac{{n(n + 1)}}{2} - n\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{n}{2}\left( {2 + (n - 1)2} \right){\text{ (using }}{S_n} = \frac{n}{2}\left( {2{u_1} + (n - 1)d} \right))\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{n}{2}(1 + 2n - 1){\text{ (using }}{S_n} = \frac{n}{2}({u_1} + {u_n}))\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>THEN</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {n^2}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({47^2} - {14^2} = 2013\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">a pentagon and five diagonals <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">five diagonals (circle optional) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Each point joins to <em>n</em> – 3 other points. <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">a correct argument for \({n(n - 3)}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">a correct argument for \(\frac{{n(n - 3)}}{2}\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) attempting to solve \(\frac{1}{2}n(n - 3) > 1\,000\,000\) for <em>n</em>. <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n > 1415.7\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n = 1416\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) <em>np</em> = 4 and <em>npq</em> = 3 <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">attempting to solve for <em>n</em> and <em>p</em> <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(n = 16\) and \(p = \frac{1}{4}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(X \sim B(16,0.25)\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(P(X = 1) = 0.0534538...( = \left( {\begin{array}{*{20}{c}}<br> {16} \\ <br> 1 <br>\end{array}} \right)(0.25){(0.75)^{15}})\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(P(X = 3) = 0.207876...( = \left( {\begin{array}{*{20}{c}}<br> {16} \\ <br> 3 <br>\end{array}} \right){(0.25)^3}{(0.75)^{13}})\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 1) + {\text{P}}(X = 3)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.261 <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;"><em>[8 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">In part (a) (i), a large number of candidates were unable to correctly use sigma notation to express the sum of the first <em>n </em>positive odd integers. Common errors included summing \(2n - 1\) from 1 to <em>n </em>and specifying sums with incorrect limits. Parts (a) (ii) and (iii) were generally well done.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (b) (i) and (iii) were generally well done. In part (b) (iii), many candidates unnecessarily simplified their quadratic when direct GDC use could have been employed. A few candidates gave \(n > 1416\) as their final answer. While some candidates displayed sound reasoning in part (b) (ii), many candidates unfortunately adopted a ‘proof by example’ approach.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (c) was generally well done. In part (c) (ii), some candidates multiplied the two probabilities rather than adding the two probabilities.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Tim goes to a popular restaurant that does not take any reservations for tables. It has </span><span style="font-family: times new roman,times; font-size: medium;">been determined that the waiting times for a table are normally distributed with a mean </span><span style="font-family: times new roman,times; font-size: medium;">of \(18\) minutes and standard deviation of \(4\) minutes.</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Tim says he will leave if he is not seated at a table within \(25\) minutes of arriving </span><span style="font-family: times new roman,times; font-size: medium;">at the restaurant. Find the probability that Tim will leave without being seated.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Tim has been waiting for \(15\) minutes. Find the probability that he will be seated </span><span style="font-family: times new roman,times; font-size: medium;">within the next five minutes.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">the waiting time, \(X{\text{ ~ }}N\)(\(18\), \({4^2}\))</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(a) \({\text{P}}(X > 25) = 0.0401\) <em><strong>(M1)A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) \({\text{P}}(X < 20|X > 15)\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{{{\text{P}}(15 < X < 20)}}{{{\text{P}}(X > 15)}}\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Only one of the above <em><strong>A1</strong></em> marks can be implied.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{{0.4648...}}{{0.7733...}} = 0.601\) <em><strong>(M1)A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was well answered, whilst few candidates managed to correctly use conditional</span> <span style="font-family: times new roman,times; font-size: medium;">probability for part (b).</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable \(X\) has a Poisson distribution with mean \(\mu \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \({\text{P}}(X = 2) + {\text{P}}(X = 3) = {\text{P}}(X = 5)\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(a) find the value of \(\mu \);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) find the probability that <em>X </em>lies within one standard deviation of the mean.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(\frac{{{\mu ^2}{{\text{e}}^{ - \mu }}}}{{2!}} + \frac{{{\mu ^3}{{\text{e}}^{ - \mu }}}}{{3!}} = \frac{{{\mu ^5}{{\text{e}}^{ - \mu }}}}{{5!}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\mu ^2}}}{2} + \frac{{{\mu ^3}}}{6} - \frac{{{\mu ^5}}}{{120}} = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\mu = 5.55\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \(\sigma = \sqrt {5.55 \ldots } = 2.35598 \ldots \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(3.19 \leqslant X \leqslant 7.9)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(4 \leqslant X \leqslant 7)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.607\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [4 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Casualties arrive at an accident unit with a mean rate of one every 10 minutes. Assume that the number of arrivals can be modelled by a Poisson distribution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the probability that there are no arrivals in a given half hour period.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) A nurse works for a two hour period. Find the probability that there are fewer than ten casualties during this period.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Six nurses work consecutive two hour periods between 8am and 8pm. Find the probability that no more than three nurses have to attend to less than ten casualties during their working period.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Calculate the time interval during which there is a 95 % chance of there being at least two casualties.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Accept exact answers in parts (a) to (c).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) number of patients in 30 minute period = <em>X</em> <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{Po(3)}}\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) number of patients in working period = <strong><em>Y</em></strong> <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(Y \sim {\text{Po(12)}}\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) number of working period with less than 10 patients = <em>W</em> <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(W \sim {\text{B}}(6,{\text{ }}0.2424 \ldots )\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica; min-height: 28.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) number of patients in t minute interval = <em>X</em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{Po}}(T)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \geqslant 2) = 0.95\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 0) + {\text{P}}(X = 1) = 0.05\) <strong><em>(M1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({{\text{e}}^{ - T}}(1 + T) = 0.05\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(T = 4.74\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>t</em> = 47.4 minutes <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [15 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Parts (a) and (b) were well answered, but many students were unable to recognise the Binomial distribution in part (c) and were unable to form the correct equation in part (d). There were many accuracy errors in this question.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A small car hire company has two cars. Each car can be hired for one whole day at a time. The rental charge is US$60 per car per day. The number of requests to hire a car for one whole day may be modelled by a Poisson distribution with mean 1.2.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that on a particular weekend, three requests are received on Saturday and none are received on Sunday.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Over a weekend of two days, it is given that a total of three requests are received.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the expected total rental income for the weekend.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(X \sim {\text{Po}}(1.2)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 3) \times {\text{P}}(X = 0)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.0867 \ldots \times 0.3011 \ldots \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.0261\) <strong><em>A1 </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Three requests over two days can occur as (3, 0), (0, 3), (2, 1) or (1, 2). <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">using conditional probability, for example</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{P}}(3,{\text{ }}0)}}{{{\text{P}}(3{\text{ requests, }}m = 2.4)}} = 0.125{\text{ or }}\frac{{{\text{P}}(2,{\text{ }}1)}}{{{\text{P}}(3{\text{ requests, }}m = 2.4)}} = 0.375\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">expected income is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2 \times 0.125 \times {\text{US}}\$ 120 + 2 \times 0.375 \times {\text{US}}\$ 180\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note:</strong><span style="font-family: 'times new roman', times; font-size: medium;"> Award </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>M1</em></strong><span style="font-family: 'times new roman', times; font-size: medium;"> for attempting to find the expected income including both (3, 0) and (2, 1) cases.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\text{US}}\$ 30 + {\text{US}}\$ 135\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = {\text{US}}\$ 165\) </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>A1</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was generally well done although a number of candidates added the two probabilities rather than multiplying the two probabilities. A number of candidates specified the required probability correct to two significant figures only.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (b) challenged most candidates with only a few candidates able to correctly employ a conditional probability argument.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Testing has shown that the volume of drink in a bottle of mineral water filled by <strong>Machine A</strong> at a bottling plant is normally distributed with a mean of \(998\) ml and a standard deviation of \(2.5\) ml.</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Show that the probability that a randomly selected bottle filled by Machine A </span><span style="font-family: times new roman,times; font-size: medium;">contains more than \(1000\) ml of mineral water is \(0.212\).<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) A random sample of \(5\) bottles is taken from Machine A. Find the probability that </span><span style="font-family: times new roman,times; font-size: medium;">exactly \(3\) of them each contain more than \(1000\) ml of mineral water.<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) Find the minimum number of bottles that would need to be sampled to ensure </span><span style="font-family: times new roman,times; font-size: medium;">that the probability of getting at least one bottle filled by Machine A containing </span><span style="font-family: times new roman,times; font-size: medium;">more than \(1000\) ml of mineral water, is greater than \(0.99\).<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(d) It has been found that for <strong>Machine B</strong> the probability of a bottle containing less </span><span style="font-family: times new roman,times; font-size: medium;">than \(996\) ml of mineral water is \(0.1151\). The probability of a bottle containing </span><span style="font-family: times new roman,times; font-size: medium;">more than \(1000\) ml is \(0.3446\). Find the mean and standard deviation for the </span><span style="font-family: times new roman,times; font-size: medium;">volume of mineral water contained in bottles filled by Machine B.<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(e) The company that makes the mineral water receives, on average, m phone calls </span><span style="font-family: times new roman,times; font-size: medium;">every \(10\) minutes. The number of phone calls, \(X\) , follows a Poisson distribution </span><span style="font-family: times new roman,times; font-size: medium;">such that \({\text{P}}(X = 2) = {\text{P}}(X = 3) + {\text{P}}(X = 4)\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) Find the value of \(m\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the probability that the company receives more than two telephone </span><span style="font-family: times new roman,times; font-size: medium;">calls in a randomly selected \(10\) minute period.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) \(X \sim {\text{N}}\)(\(998\), \({2.5^2}\) ) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(X > 1000) = 0.212\) <em><strong>AG</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) \(X \sim {\text{B}}\)(\(5\), \(0.2119...\))</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of binomial <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}(X = 3) = \left( {\begin{array}{*{20}{c}}<br> 5 \\ <br> 3 <br>\end{array}} \right){\left( {0.2119...} \right)^3}{\left( {0.7881...} \right)^2} = 0.0591\) </span><span style="font-family: times new roman,times; font-size: medium;"> (accept \(0.0592\)) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)A1</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) \({\text{P}}(X \geqslant 1) = 1 - {\text{P}}(X = 0)\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(1 - {\left( {0.7881...} \right)^n} > 0.99\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\left( {0.7881...} \right)^n} < 0.01\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Award <em><strong>A1</strong></em> for line 2 or line 3 or equivalent.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(n > 19.3\) <em><strong>(A1)</strong></em></span><br><span style="font-family: times new roman,times; font-size: medium;">minimum number of bottles required is \(20\) <em><strong>A1N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(d) </span><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{996 - \mu }}{\sigma } = - 1.1998\) </span><span style="font-family: times new roman,times; font-size: medium;"> (accept \(1.2\)) <em><strong>M1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{1000 - \mu }}{\sigma } = 0.3999\) </span><span style="font-family: times new roman,times; font-size: medium;"> (accept \(0.4\))</span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>M1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\mu = 999\)(<em>ml</em>), \(\sigma = 2.50\)(<em>ml</em>) <em><strong>A1A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(e) (i) </span><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{{{\text{e}}^{ - m}}{m^2}}}{{2!}} = \frac{{{{\text{e}}^{ - m}}{m^3}}}{{3!}} + \frac{{{{\text{e}}^{ - m}}{m^4}}}{{4!}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{{m^2}}}{2} = \frac{{{m^3}}}{6} + \frac{{{m^4}}}{{24}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(12{m^2} - 4{m^3} - {m^4} = 0\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(m = - 6\), \(0\), \(2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow m = 2\) <em><strong>A1N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({\text{P}}(X > 2) = 1 - {\text{P}}(X \leqslant 2)\) <em><strong>(M1)</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = 1 - {\text{P}}(X = 0) - {\text{P}}(X = 1) - {\text{P}}(X = 2)\)</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = 1 - {{\text{e}}^{ - 2}} - 2{{\text{e}}^{ - 2}} - \frac{{{2^2}{{\text{e}}^{ - 2}}}}{{2!}}\)</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = 0.323\) <em><strong>A1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">Total [20 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">This was the best done of the section B questions, with the majority of candidates making the correct choice of probability distribution for each part. The main sources of errors: (b) missing out the binomial coefficient in the calculation; (c) failure to rearrange 'at least one bottle' in terms of the probability of obtaining no bottles; (d) using \(1.2\) rather than \( - 1.2\) in the inverse Normal or not performing an inverse Normal at all; (e)(ii) misinterpreting 'more than two'.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The lifts in the office buildings of a small city have occasional breakdowns. The breakdowns at any given time are independent of one another and can be modelled using a Poisson Distribution with mean 0.2 per day.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Determine the probability that there will be exactly four breakdowns during the month of June (June has 30 days).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Determine the probability that there are more than 3 breakdowns during the month of June.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Determine the probability that there are no breakdowns during the first five days of June.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Find the probability that the first breakdown in June occurs on June \({3^{{\text{rd}}}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) It costs 1850 Euros to service the lifts when they have breakdowns. Find the expected cost of servicing lifts for the month of June.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) Determine the probability that there will be no breakdowns in exactly 4 out of the first 5 days in June.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) mean for 30 days: \(30 \times 0.2 = 6\) . <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 4) = \frac{{{6^4}}}{{4!}}{{\text{e}}^{ - 6}} = 0.134\) <strong><em>(M1)A1</em></strong> <strong><em>N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) \({\text{P}}(X > 3) = 1 - {\text{P}}(X \leqslant 3) = 1 - {{\text{e}}^{ - 6}}(1 + 6 + 18 + 36) = 0.849\) <strong><em>(M1)A1</em></strong> <strong><em>N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">mean for five days: \(5 \times 0.2 = 1\) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 0) = {{\text{e}}^{ - 1}}\,\,\,\,\,( = 0.368)\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">mean for one day: 0.2 <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X = 0) = {({{\text{e}}^{ - 0.2}})^5} = {{\text{e}}^{ - 1}}\,\,\,\,\,( = 0.368)\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Required probability \( = {{\text{e}}^{ - 0.2}} \times {{\text{e}}^{ - 0.2}} \times (1 - {{\text{e}}^{ - 0.2}})\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">= 0.122 <strong><em>A1</em></strong> <strong><em>N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(e) Expected cost is \(1850 \times 6 = {\text{11}}\,{\text{100 Euros}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(f) On any one day \({\text{P}}(X = 0) = {{\text{e}}^{ - 0.2}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Therefore, \(\left( {\begin{array}{*{20}{c}}<br> 5 \\ <br> 1 <br>\end{array}} \right){({{\text{e}}^{ - 0.2}})^4}(1 - {{\text{e}}^{ - 0.2}}) = 0.407\) <strong><em>M1A1</em></strong> <strong><em>N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [13 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Many candidates showed familiarity with the Poisson Distribution. Parts (a), (b), and (c) were straightforward, as long as candidates multiplied 0.2 by 30 to get the mean. Part (e) was answered successfully by most candidates. Parts (d) and (f) were done very poorly. In part (d), most candidates calculated \({\text{P}}(X = 1)\) rather than \({\text{P}}(X \leqslant 1)\). Although some candidates realized the need for the Binomial in part (e), some incorrectly used 0.8 and 0.2.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The wingspans of a certain species of bird can be modelled by a normal distribution with mean \(60.2\) cm and standard deviation \(2.4\) cm.</p>
<p class="p1">According to this model, \(99\% \) of wingspans are greater than <em>\(x\) </em>cm.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(x\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">In a field experiment, a research team studies a large sample of these birds. The wingspans of each bird are measured correct to the nearest \(0.1\) cm.</p>
<p class="p1">Find the probability that a randomly selected bird has a wingspan measured as \(60.2\) cm.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P}}(X > x) = 0.99\;\;\;\left( { = {\text{P}}(X < x) = 0.01} \right)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1">\( \Rightarrow x = 54.6{\text{ (cm)}}\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\({\text{P}}(60.15 \le X \le 60.25)\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)(A1)</em></strong></span></p>
<p class="p1">\( = 0.0166\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Many candidates did not use the symmetry of the normal curve correctly. Many, for example, calculated the value of \(x\) for which \({\text{P}}(X < x) = 0.99\) rather than \({\text{P}}(X < x) = 0.01\).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Most candidates did not recognize that the required probability interval was \({\text{P}}(60.15 \le X \le 60.25)\). A large number of candidates simply stated that \({\text{P}}(X = 60.2) = 0.166\). Some candidates used \({\text{P}}(60.1 \le X \le 60.3)\) while a number of candidates bizarrely used probability intervals not centred on \(60.2\), for example, \({\text{P}}(60.15 \le X \le 60.24)\).</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the term in \({x^5}\) in the expansion of \((3x + A){(2x + B)^6}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span style="font-family: 'times new roman', times; font-size: medium;">Mina and Norbert each have a fair cubical die with faces labelled 1, 2, 3, 4, 5 and 6; they throw</span></p>
<p class="p1"><span style="font-family: 'times new roman', times; font-size: medium;">it to decide if they are going to eat a cookie.</span></p>
<p class="p2"><span style="font-family: 'times new roman', times; font-size: medium;">Mina throws her die just once and she eats a cookie if she throws a four, a five or a six.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="line-height: 20px; background-color: #f7f7f7;">Norbert throws his die six times and each time eats a cookie if he throws a five or a six.</span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the probability that five cookies are eaten.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( {A\left( \begin{array}{l}6\\5\end{array} \right){2^5}B + 3\left( \begin{array}{l}6\\4\end{array} \right){2^4}{B^2}} \right){x^5}\) <strong><em>M1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \left( {192AB + 720{B^2}} \right){x^5}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \frac{1}{6},{\text{ }}A = \frac{3}{6}\left( { = \frac{1}{2}} \right),{\text{ }}B = \frac{4}{6}\left( { = \frac{2}{3}} \right)\) <strong><em>A1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">probability is \(\frac{4}{{81}}{\text{ }}( = 0.0494)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">P (5 eaten) =P (M eats 1) P (N eats 4) + P (M eats 0) P (N eats 5) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{2}\left( \begin{array}{l}6\\4\end{array} \right){\left( {\frac{1}{3}} \right)^4}{\left( {\frac{2}{3}} \right)^2} + \frac{1}{2}\left( \begin{array}{l}6\\5\end{array} \right){\left( {\frac{1}{3}} \right)^5}\left( {\frac{2}{3}} \right)\) <strong><em>(A1)(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{4}{{81}}{\text{ }}( = 0.0494)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Consider the data set \(\{ 2,{\text{ }}x,{\text{ }}y,{\text{ }}10,{\text{ }}17\} ,{\text{ }}x,{\text{ }}y \in {\mathbb{Z}^ + }\) and \(x < y\).</p>
<p class="p1">The mean of the data set is \(8\) and its variance is \(27.6\).</p>
<p class="p1">Find the value of \(x\) and the value of \(y\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>use of \(\mu = \frac{{\sum\limits_{i = 1}^k {{f_i}{x_i}} }}{n}{\text{ }}\) to obtain \(\frac{{2 + x + y + 10 + 17}}{5} = 8\) <strong><em>(M1)</em></strong></p>
<p>\(x + y = 11\) <strong><em>A1</em></strong></p>
<p><strong>EITHER</strong></p>
<p>use of \({\sigma ^2} = \frac{{\sum\limits_{i = 1}^k {{f_i}{{({x_i} - \mu )}^2}} }}{n}\) to obtain \(\frac{{{{( - 6)}^2} + {{(x - 8)}^2} + {{(y - 8)}^2} + {2^2} + {9^2}}}{5} = 27.6\) <strong><em>(M1)</em></strong></p>
<p>\({(x - 8)^2} + {(y - 8)^2} = 17\) <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p>use of \({\sigma ^2} = \frac{{\sum\limits_{i = 1}^k {{f_i}x_i^2} }}{n} - {\mu ^2}\) to obtain \(\frac{{{2^2} + {x^2} + {y^2} + {{10}^2} + {{17}^2}}}{5} - {8^2} = 27.6\) <strong><em>(M1)</em></strong></p>
<p>\({x^2} + {y^2} = 65\) <strong><em>A1</em></strong></p>
<p><strong>THEN</strong></p>
<p>attempting to solve the two equations <strong><em>(M1)</em></strong></p>
<p>\(x = 4\;\;\;\)and\(\;\;\;y = 7\;\;\;({\text{only as }}x < y)\)<strong><em>A1 N4</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A0</em></strong> for \(x = 7\) and \(y = 4\).</p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1)A1(M0)A0(M1)A1</em></strong> for \(x + y = 11 \Rightarrow x = 4\) and \(y = 7\).</p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Reasonably well done. Most candidates were able to obtain \(x + y = 11\). Most manipulation errors occurred when candidates attempted to form the variance equation in terms of \(x\) and \(y\). Some candidates did not apply the condition \(x < y\) when determining their final answer.</p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A discrete random variable <em>X</em> has a probability distribution given in the following table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0px; font: 27px Helvetica; text-align: center;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \({\text{E}}(X) = 2.61\), determine the value of <em>p</em> and of <em>q</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Calculate \({\text{Var}}(X)\) to three significant figures.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) \(p + q = 0.44\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(2.5p + 3.5q = 1.25\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(p = 0.29,{\text{ }}q = 0.15\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) use of \({\text{Var}}(X) = {\text{E}}({X^2}) - {\text{E}}{(X)^2}\)</span><span style="font-family: 'times new roman', times; font-size: medium;"> </span><strong style="font-family: 'times new roman', times; font-size: medium;"><em>(M1)</em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(X) = 2.10\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">An easy question, well answered by most candidates. For the others it was disappointing that many did not use the fact that the probabilities add to unity.</span></p>
</div>
<br><hr><br><div class="specification">
<p>The number of bananas that Lucca eats during any particular day follows a Poisson distribution with mean 0.2.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that Lucca eats at least one banana in a particular day.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the expected number of weeks in the year in which Lucca eats no bananas.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>let \(X\) be the number of bananas eaten in one day</p>
<p>\(X \sim {\text{Po}}(0.2)\)</p>
<p>\({\text{P}}(X \geqslant 1) = 1 - {\text{P}}(X = 0)\) <strong><em>(M1)</em></strong></p>
<p>\( = 0.181{\text{ }}( = 1 - {{\text{e}}^{ - 0.2}})\) <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>let \(Y\) be the number of bananas eaten in one week</p>
<p>\({\text{Y}} \sim {\text{Po}}(1.4)\) <strong><em>(A1)</em></strong></p>
<p>\({\text{P}}(Y = 0) = 0.246596 \ldots {\text{ }}( = {{\text{e}}^{ - 1.4}})\) <strong><em>(A1)</em></strong></p>
<p><strong>OR</strong></p>
<p>let \(Z\) be the number of days in one week at least one banana is eaten</p>
<p>\(Z \sim {\text{B}}(7,{\text{ }}0.181 \ldots )\) <strong><em>(A1)</em></strong></p>
<p>\({\text{P}}(Z = 0) = 0.246596 \ldots \) <strong><em>(A1)</em></strong></p>
<p><strong>THEN</strong></p>
<p>\(52 \times 0.246596 \ldots \) <strong><em>(M1)</em></strong></p>
<p>\( = 12.8{\text{ }}( = 52{{\text{e}}^{ - 1.4}})\) <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">After being sprayed with a weedkiller, the survival time of weeds in a field is normally distributed with a mean of 15 days.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) If the probability of survival after 21 days is 0.2 , find the standard deviation of the survival time.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">When another field is sprayed, the survival time of weeds is normally distributed with a mean of 18 days.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) If the standard deviation of the survival time is unchanged, find the probability of survival after 21 days.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) required to solve \({\text{P}}\left( {Z < \frac{{21 - 15}}{\sigma }} \right) = 0.8\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{6}{\sigma } = 0.842 \ldots \,\,\,\,\,\)(or equivalent) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( \Rightarrow \sigma = 7.13\) (days) <strong><em>A1 N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) P(survival after 21 days) = 0.337 <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[5 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A straightforward Normal distribution problem, but many candidates confused the <em>z</em> value with the probability.</span></p>
</div>
<br><hr><br><div class="specification">
<p>The random variable <em>X</em> has a normal distribution with mean <em>μ</em> = 50 and variance <em>σ </em><sup>2</sup> = 16 .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the probability density function for<em> X</em>, and shade the region representing P(<em>μ</em> − 2σ < <em>X</em> < <em>μ</em> + σ).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of P(<em>μ</em> − 2σ < <em>X</em> < <em>μ</em> + σ).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>k</em> for which P(<em>μ</em> − <em>k</em>σ < <em>X</em> < <em>μ</em> + <em>k</em>σ) = 0.5.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img 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"></p>
<p>normal curve centred on 50Â Â Â <em><strong>A1</strong></em></p>
<p>vertical lines at \(x\) = 42 and \(x\) = 54, with shading in between   <em><strong>  A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>P(42<em> </em>< <em>X</em> < 54) (= P(− 2<em> </em>< <em>Z</em> < 1))   <em><strong>(M1)</strong></em></p>
<p>= 0.819Â Â Â <em><strong> Â A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>P(<em>μ</em> − <em>k</em>σ < <em>X</em> < <em>μ</em> + <em>k</em>σ) = 0.5 ⇒ P(<em>X</em> < <em>μ</em> + <em>k</em>σ) = 0.75    <em><strong>(M1)</strong></em></p>
<p><em>k</em>Â = 0.674Â Â Â <em><strong> Â A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong></em> for <em>k</em> = −0.674<em>.</em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Tim throws two identical fair dice simultaneously. Each die has six faces: two faces numbered 1, two faces numbered 2 and two faces numbered 3. His score is the sum of the two numbers shown on the dice.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Calculate the probability that Tim obtains a score of 6.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 27px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Calculate the probability that Tim obtains a score of at least 3.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Tim plays a game with his friend Bill, who also has two dice numbered in the same way. Bill’s score is the sum of the two numbers shown on his dice.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Calculate the probability that Tim and Bill <strong>both</strong> obtain a score of 6.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 27px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Calculate the probability that Tim and Bill obtain the same score.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Let <em>X</em> denote the largest number shown on the four dice.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 27px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) Show that \({\text{P}}(X \leqslant 2) = \frac{{16}}{{81}}\).</span></p>
<p style="margin: 0px 0px 0px 30px; font: 27px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Copy and complete the following probability distribution table.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 27px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<p style="margin: 0px 0px 0px 30px; font: 27px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0px 0px 0px 30px; font: 27px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (iii) Calculate \({\text{E}}(X)\) and \({\text{E}}({X^2})\) and hence find \({\text{Var}}(X)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Given that <em>X</em> = 3, find the probability that the sum of the numbers shown on the four dice is 8.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) let <em>T</em> be Tim’s score</span></p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; font: 27px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\text{P}}(T = 6) = \frac{{11}}{9}{\text{ }}( = 0.111{\text{ 3 sf)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; font: 27px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; font: 27px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\text{P}}(T \geqslant 3) = 1 - {\text{P}}(T \leqslant 2) = 1 - \frac{1}{9} = \frac{8}{9}{\text{ }}( = 0.889{\text{ 3 sf)}}\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) let <em>B</em> be Bill’s score</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \({\text{P}}(T = 6{\text{ and }}B = 6) = \frac{1}{9} \times \frac{1}{9} = \frac{1}{{81}}{\text{ }}( = 0.012{\text{ 3 sf)}}\) <strong><em>(M1)A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \({\text{P}}(B = T) = {\text{P}}(2){\text{P}}(2) + {\text{P}}(3){\text{P}}(3) + \ldots + {\text{P}}(6){\text{P}}(6)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{9} \times \frac{1}{9} + \frac{2}{9} \times \frac{2}{9} + \frac{3}{9} \times \frac{3}{9} + \frac{2}{9} \times \frac{2}{9} + \frac{1}{9} \times \frac{1}{9}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{19}}{{81}}{\text{ }}( = 0.235{\text{ 3 sf)}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) <strong>EITHER</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \leqslant 2) = \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">because \({\text{P}}(X \leqslant 2) = {\text{P}}\left( {(a,{\text{ }}b,{\text{ }}c,{\text{ }}d)|a,{\text{ }}b,{\text{ }}c,{\text{ }}d = 1,{\text{ }}2} \right)\) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">or equivalent</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \leqslant 2) = \frac{{16}}{{81}}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>OR</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">there are sixteen possible permutations which are</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><img src="data:image/png;base64,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" alt> <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> This information may be presented in a variety of forms.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(X \leqslant 2) = \frac{{1 + 4 + 6 + 4 + 1}}{{81}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{16}}{{81}}\) <strong><em>AG</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) </span><span style="font-family: 'times new roman', times; font-size: medium;"><img src="data:image/png;base64,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" alt> <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) \({\text{E}}(X) = \sum\limits_{x = 1}^3 {x{\text{P}}(X = x)} \) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{{81}} + \frac{{30}}{{81}} + \frac{{195}}{{81}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{226}}{{81}}\,\,\,\,\,\)(2.79 to 3 sf) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{E}}({X^2}) = \sum\limits_{x = 1}^3 {{x^2}{\text{P}}(X = x)} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{1}{{81}} + \frac{{60}}{{81}} + \frac{{585}}{{81}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{646}}{{81}}\,\,\,\,\,\)(7.98 to 3 sf) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{Var}}(X) = {\text{E}}({X^2}) - {\left( {{\text{E}}(X)} \right)^2}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.191\) (3 sf) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>M1A0</em></strong> for answers obtained using rounded values \(\left( {e.g.{\text{ Var}}(X) = 0.196} \right)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[10 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) </span><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQwAAAA9CAIAAADXv8B4AAAJLUlEQVR4nO2dv2sjORTH9W+odusu3VVTuUmzTZoUKtykSHfFgItAioMtBgwLCwsLA2HhILAIAsvBYRCBYxvDsM1ymIErQhjUhCMMUxhjxLtC0vzyjDe7OzO2uPfp7LFlW1999SSNnkwIgiDfBBA3Qe0GAE3iNqjdAKBJ3Aa1GwA0idugdgOAJnEb1G4AOjeJyuL7jwGjulzKgo9/RHKd/nX/ZfvzhW8lv9AFj4JotzyV8CkldMoT9VMfoxJx9yXrtsye6MUkkrN8SYdOw1Vqnt9GwchUP+OPXXzSNwQ9Ejo1iZLLOaOEEM+/iaQCAFBZ/GfAxt3VwSNno35Noh749FUQ/Y9NAlBuvoRe8mRjn3+O5q9LD3+efYIeCR2aJF2FU0oI8a6FrFZitgxO5wOYpANUImYTQrzcJEdOnyaZB+HM0+3Dm0eZMs9/+tRp1f+fTKJ7XELoJFzt9LnKDrfSWIT+xPN+oYRM/FDEmYJyGGe/R8u3jBJCxmz+WaZfQzauBn1bp374hx7UeTMep5VCRkG0LT08D/iN7xFCyGQmbDxQMuL5oHDi3yylAshW3J+U7h5d3PLfymUCqCwWof/K804IoZ4fijiFymefBzz0PUoI9WYLHUyrxV5w2WVj6NMk77j813R8JI+luUmqI6WiBi64LNf+xcfofs7GhBDK3i7lky1wzMKvGcAeQTVKLm907dnhSam18ER+nrMxoTOR9hjouzJJXmV7+mCVRXOPEMK4rA9jsijwcjNYvxFCqDe7E+/PiH1XXqeU3awyZV5pxgPrODwrNei8zEuerFMxo8Wl5yg4JeQsjNeQCp8SQk588QSQK5D/imqZ2TLwqGkH6oFPx/lQxCo3nvIHZcrUhazj8MyomMWL4M2dSybZ2tBKCKFesMwqkeRJ+Cd53dgaML1AXiEs/JrpujJB6Xoh3k6KV+4R1Fa4N4+ydcIvaU3o8+DuS5KubtjIDZPYFrnPJOa32cCqa+csjNeFx7QTTAXrztheMm+rRWetkw5ftVfar8S4rEuoR4Z0Eq6UuWBnonWTVMpUcTgpyteXTOSslF8pU//qMZt/lj3o2LtJACCz8ZyMp/yfpDCJqeFGk9jZv35otdBDcXNJV/IeQSsNZhsFo2ahe6crk1hz511yA5U6tbWjW1KjSXQl7jdJucyXm0SzkRGfv748py81iW38QWW8sVt+1XhKLmYeJaQ+kOiEIUxS/gnEYxfhD5vE1Nw+k5Rfmfe8BdQXaa219E9XJrFDqeY5iaY/k3xvJLHzBG8ePS9fHkl+zCQAAFm8MDOg0yB6/unaLhjGJAAqW92YKVzRf/dnEi2oGc5RX6T17+akSQDycWexEpJfSsTNvQSlJwbmN6tVOKF1J3y3SfRDHb5ebpJSTP+e4ZaZwJhp4u5goN0kAHk763aMMJRJoOgH+zVJWVA7PJmEsap9N0dNAqD0UgMhlM0XZlyhsliEsyuzNlXMw1S1wdkGraujMpmuTcftPG/KE1Wb/dtX1sqsmOQsjNfFnCdYpnqakQdAc+nEF0ly//kfVStTz/h1NNBOM8PLJpPY+Hb3JljEmfn5J1P+0OHcpDeTrOPQ3wl6m4Rf0sIkpnKoL9Jifl+buOs5ZyUmmKld1SRNguZLpmYpTMnoUyRVrbX0T5cmAQCANBa3AbNLGWTih5+i0m0Tu6I3PmevqFkwKS3qETIKbm9Z/qi0CGts8sj9a75aRWZVd8yCP+vryIQQxh+LhyN2e1tcY1zaWzqUBTz6exmcErNAuQHYSL0GTafhKq2VKQFASf3R9JydUxs2y69jIS9G0hdcruX9X6tEdx/dT997MUnp5+zEvefozYfiqXxO781uw1/zlVlZUfQ3fnuRP6hoMQqibaugAACgspj7ejpEme5rqq1liKl75yZBBgW1GwA0idugdgOAJnEb1G4ACIIgL+LQXkV+ENRuANAkboPaDQCaxG1QuwFAk7gNajcAaBK3Qe0GwDGT5Dtfdm7N2jvlpLg1m78pi+956Hu7CU9ZLPh73xt3lLF9ABzSrnr+QcfJZ73ilEmy5dx/ZzaPiGuv2HGssuidr3d8mE1EessQAOQ7wXaFMRuKujvW4AA4ox2kMZ95hHr++7uoj8yaHnHKJBUeORvtbKIGAL3JtJbWorfiNfReKg4naJLe2ST8khYpu47hrElS4Y8um47tUKmYjeqHm+jN1U0hXnKGJumZIz9x5pu4aBJ9TNGkKQUyjRfzpoxnNMkBsQnMr6+Z2c6rN1w7g3MmKad0jiu5GeUt1LQWZNAkh0Nn15mzTuz6ylCpIJ3gnEkAoHQg0M5ZMsWVynQFTXI49AkOhRylo2ocwU2TAOyZi5uuq5LbiSY5HHWTaC2cOf4PXDaJSsWM7ubTAwA8Cf8XnfNoQZMckCfhn5Ri/lbyi76Pk+sWV02i5GLmnRYnMhZspLj26ket6rDTEOJVHE72nfBy7DihnUr4lJojLZVczLwTt9aCHTJJfmoRIfV7UnqYq6ln1VeT38tZ0dVjnY75MNp2nNEuT1XPD5V1B4dMgjSA2g0AmsRtULsBQJO4DWo3AARBkBdxaK8iPwhqNwBoErdB7QYATeI2qN0AoEncBrUbADSJ26B2A+CYSV6U4169p1t6C/X8D5Wb8Zjj3iNtRwu0KnW0OGWSfTnub5jP40zZ/5I3f+oA2TJgVzxOC22KPZGY494nzUcLtCt1xDhlkgrlHPcnEbwvtgPrrfKjINqqVLydF/9Eo7c5VjZpY457n+ymM7QpdaAv+DKcNUlrjju0nxHRtEkbt8r3SHuGgqH9NI9jwkWT7MlxBwAd6Bv/BPhJ+Cd1SdAkPfItk7QqdVw4Z5L2HHfDJuGXO6elAOisht3ggybpkf0maVXq2HDOJACwN8c94dNJU7qieuDT84a/h0aT9Mg+k7QqdXy4aRKA5hz37GvoB6LhuJp0FV5dNaQxokl6pd0krUodI+6aZCfHXSXiyv4XdoWNFIHfli+KJumRFpO0KnWkuGqSeo57vd43UswD8bTrECUXV8F9/jrMce+TpqMFWpU6XhwySXuOuzkkuwqdiXQtxbVXv5Avp2COe580Hi3QqtRR91EOmQRpALUbADSJ26B2A4AmcRvUbgDq40MEQXb5DwaSbkFJfhfkAAAAAElFTkSuQmCC" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}\left( {{\text{total is }}8 \cap (X = 3)} \right) = \frac{{18}}{{81}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">since \({\text{P}}(X = 3) = \frac{{65}}{{81}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}\left( {{\text{total is }}8|(X = 3)} \right) = \frac{{{\text{P}}\left( {({\text{total is 8)}} \cap (X = 3)} \right)}}{{{\text{P}}(X = 3)}}\) <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{18}}{{65}}{\text{ }}( = 0.277)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>Total [21 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Most candidates with a reasonable understanding of probability managed to answer well parts (a), (b) and some of part (c). However some candidates did not realize that different scores were not equally likely which lead to incorrect answers in several parts. Surprisingly, many candidates completed the table in part c) ii) with values that did not add up to 1. Very few candidates answered part (d) well. The enumeration of possible cases was sometimes attempted but with little success.</span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">In each round of two different games Ying tosses three fair coins and Mario tosses two fair coins.</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) The first game consists of one round. If Ying obtains more heads than Mario, she receives $5 from Mario. If Mario obtains more heads than Ying, he receives $10 from Ying. If they obtain the same number of heads, then Mario receives $2 from Ying. Determine Ying’s expected winnings.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) They now play the second game, where the winner will be the player who obtains the larger number of heads in a round. If they obtain the same number of heads, they play another round until there is a winner. Calculate the probability that Ying wins the game.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Ying:</span></p>
<p style="margin-left: 30px;"><img src="data:image/png;base64,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" alt> <em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Mario:</span></p>
<p style="margin-left: 30px;"><img src="data:image/png;base64,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" alt><span style="font-family: times new roman,times; font-size: medium;"> </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(Ying wins)}} = \frac{1}{8} + \frac{3}{8}\left( {\frac{2}{4} + \frac{1}{4}} \right) + \frac{3}{8} \times \frac{1}{4}\)</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{{16}}{{32}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(Mario wins)}} = \frac{1}{4}\left( {\frac{3}{8} + \frac{1}{8}} \right) + \frac{2}{4} \times \frac{1}{8}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{6}{{32}}\) <em><strong>(M1)A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(draw)}} = 1 - \frac{{16}}{{32}} - \frac{6}{{32}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{{10}}{{32}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Ying’s winnings:</span></p>
<p style="margin-left: 30px;"><img src="data:image/png;base64,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" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;">expected winnings \( = 5\left( {\frac{{16}}{{32}}} \right) - 105\left( {\frac{6}{{32}}} \right) - 25\left( {\frac{{10}}{{32}}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = 0\) <em><strong>A1</strong></em></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[12 marks]</span></em></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) \({\text{P(Ying wins on 1st round)}} = \frac{1}{2}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(Ying wins on 2st round)}} = \frac{5}{{16}} \times \frac{1}{2}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)(A1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(Ying wins on 3rd round)}} = {\left( {\frac{5}{{16}}} \right)^2} \times \frac{1}{2}\) </span><span style="font-family: times new roman,times; font-size: medium;"><em>etc.</em> <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{P(Ying wins)}} = \frac{1}{2} + \frac{5}{{16}} \times \frac{1}{2} + {\left( {\frac{5}{{16}}} \right)^2} \times \frac{1}{2} + ...\) <em><strong>(M1)</strong></em></span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{{\frac{1}{2}}}{{1 - \frac{5}{{16}}}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{8}{{11}}\) </span><span style="font-family: times new roman,times; font-size: medium;">( \( =0.727\)) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[8 marks]</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">Total [20 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">There were some good attempts at this question, but there were also many candidates that were unable to maintain a clearly presented solution and consequently were unable to obtain marks that they should have been able to secure. Those that attempted part (b) usually made a good attempt.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Only two international airlines fly daily into an airport. UN Air has 70 flights a day and IS Air has 65 flights a day. Passengers flying with UN Air have an 18 % probability of losing their luggage and passengers flying with IS Air have a 23 % probability of losing their luggage. You overhear someone in the airport complain about her luggage being lost.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that she travelled with IS Air.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><img src="data:image/png;base64,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" alt><span style="font-family: 'times new roman', times; font-size: medium;"> <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let P(<em>I</em>) be the probability of flying IS Air, P(<em>U</em>) be the probability flying UN Air and P(<em>L</em>) be the probability of luggage lost.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(I|L) = \frac{{{\text{P}}(I \cap L)}}{{{\text{P}}(L)}}{\text{ }}\left( {{\text{or Bayes' formula , P}}(I|L) = \frac{{{\text{P}}(L|I){\text{P}}(I)}}{{{\text{P}}(L|I){\text{P}}(I) + {\text{P}}(L|U){\text{P}}(U)}}} \right)\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{0.23 \times \frac{{65}}{{135}}}}{{0.18 \times \frac{{70}}{{135}} + 0.23 \times \frac{{65}}{{135}}}}\) <strong><em>A1A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{299}}{{551}}{\text{ }}( = 0.543,{\text{ accept }}0.542)\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span><strong style="font-family: 'times new roman', times; font-size: medium;"><em> </em></strong></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Expected number of suitcases lost by UN Air is \(0.18 \times 70 = 12.6\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Expected number of suitcases lost by IS Air is \(0.23 \times 65 = 14.95\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P}}(I|L) = \frac{{14.95}}{{12.6 + 14.95}}\) <strong><em>M1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.543\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This question was well answered by the majority of candidates. Most candidates used either tree diagrams or expected value methods.</span></p>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Find the percentage of the population that has been vaccinated.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) A randomly chosen person catches the virus. Find the probability that this person </span><span style="font-family: times new roman,times; font-size: medium;">has been vaccinated.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a)</span></p>
<p><img 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" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;">using the law of total probabilities: <em><strong>(</strong><strong>M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.1p + 0.3\left( {1 - p} \right) = 0.22\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.1p + 0.3 - 0.3p = 0.22\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(0.2p = 0.88\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = \frac{{0.88}}{{0.2}} = 0.4\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 40\% \) (accept \(0.4\)) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) required probability </span><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{{0.4 \times 0.1}}{{0.22}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> M1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( = \frac{2}{{11}}\) </span><span style="font-family: times new roman,times; font-size: medium;">(\(0.182\)) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates who successfully answered this question had first drawn a tree diagram, using a symbol to denote the probability that a randomly chosen person had received the influenza virus. For those who did not draw a tree diagram, there was poor understanding of how to apply the conditional probability formula.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Kathy plays a computer game in which she has to find the path through a maze within a certain time. The first time she attempts the game, the probability of success is known to be 0.75. In subsequent attempts, if Kathy is successful, the difficulty increases and the probability of success is half the probability of success on the previous attempt. However, if she is unsuccessful, the probability of success remains the same. Kathy plays the game three times consecutively.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that she is successful in all three games.<br></span></p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Assuming that she is successful in the first game, find the probability that she is successful in exactly two games.<br></span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Times; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{P(WWW)}} = 0.75 \times 0.375 \times 0.1875 = 0.0527{\text{ (3sf) }}\left( {\frac{3}{4} \times \frac{3}{8} \times \frac{3}{{16}} = \frac{{27}}{{512}}} \right)\) <strong><em>(M1)A1</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><img 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" alt> <span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>(M1)(A1)</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong> </strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>M1 </em></strong>for any reasonable attempt to use a tree diagram showing that three games were played (do not award <strong><em>M1 </em></strong>for tree diagrams that only show the first two games) and <strong><em>A1 </em></strong>for the highlighted probabilities.</span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"> </p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11.5px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left. {{\text{P(wins 2 games}}\,} \right|{\text{wins first game)}} = \frac{{{\text{P(WWL, WLW)}}}}{{{\text{P(wins first game)}}}}\) <strong>(<em>M1)</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11.5px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = \frac{{0.75 \times 0.375 \times 0.8125 + 0.75 \times 0.625 \times 0.375}}{{0.75}}\) <strong><em>(A1)(A1)</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\( = 0.539{\text{ (3sf) }}\left( {{\text{or }}\frac{{69}}{{128}}} \right)\) <strong><em>A1</em></strong></span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 12px/normal Helvetica; margin: 0px;"><strong style="font-family: 'times new roman', times; font-size: medium;">Note: </strong><span style="font-family: 'times new roman', times; font-size: medium;">Candidates may use the tree diagram to obtain the answer without using the conditional probability formula, </span><em style="font-family: 'times new roman', times; font-size: medium;">ie</em><span style="font-family: 'times new roman', times; font-size: medium;">,</span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11.5px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\left. {{\text{P(wins 2 games}}\,} \right|{\text{wins first game)}} = 0.375 \times 0.8125 + 0.625 \times 0.375 = 0.539.\)</span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11.5px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font-family: 'Helvetica Neue', Arial, 'Lucida Grande', 'Lucida Sans Unicode', sans-serif; font: normal normal normal 11px/normal Helvetica; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was generally successful to most candidates; however the conditional probability was proved difficult to many candidates either because the unconditional probability of two correct games was found or the success in the second and third game was included. Many candidates used a clear tree diagram to calculate the corresponding probabilities. However other candidates frequently tried to do the problem without drawing a tree diagram and often had incorrect probabilities. It was sad to read many answers with probabilities greater than 1.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 10.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">Part (a) was generally successful to most candidates; however the conditional probability was proved difficult to many candidates either because the unconditional probability of two correct games was found or the success in the second and third game was included. Many candidates used a clear tree diagram to calculate the corresponding probabilities. However other candidates frequently tried to do the problem without drawing a tree diagram and often had incorrect probabilities. It was sad to read many answers with probabilities greater than 1.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times;">In a class of \(20\) students, \(12\) study Biology, \(15\) study History and \(2\) students study neither Biology nor History.</span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) Illustrate this information on a Venn diagram.<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) Find the probability that a randomly selected student from this class is studying </span><span style="font-family: times new roman,times; font-size: medium;">both Biology and History.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) Given that a randomly selected student studies Biology, find the probability that </span><span style="font-family: times new roman,times; font-size: medium;">this student also studies History.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">(a)</span></p>
<p><img 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" alt> <em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Award <em><strong>A1</strong></em> for a diagram with two intersecting regions and at least </span><span style="font-family: times new roman,times; font-size: medium;">the value of the intersection.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(b) \(\frac{9}{{20}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(c) \(\frac{9}{{12}}\left( { = \frac{3}{4}} \right)\) <em><strong>A1</strong></em><br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[4 marks]<br></strong></em></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Although this was the best done question on the paper, it was disappointing that a significant </span><span style="font-family: times new roman,times; font-size: medium;">number of candidates produced Venn diagrams with key information missing.</span></p>
</div>
<br><hr><br>