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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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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 \(T\) 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"> </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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<br><hr><br><div class="specification">
<p class="p1">A discrete random variable \(X\) follows a Poisson distribution \({\text{Po}}(\mu )\).</p>
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<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>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Given that \({\text{P}}(X = 2) = 0.241667\) </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>
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<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>
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<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>
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<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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<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>
</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;">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>
<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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<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>
<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>
<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"> </span>Hence show that \(X\) has two modes \({m_1}\) and \({m_2}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<br><hr><br>