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href="../2407/410-spearmans-rank-correlation-coefficient.html">4.10 Spearman's Rank Correlation Coefficient</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2259/411-chi-squared-independence-tests.html">4.11 Chi Squared Independence tests</a></label></li><li class=""><label style="padding-left: 0px"><i class="fa fa-fw"></i><a href="../2413/411-chi-squared-goodness-of-fit-and-the-t-test.html">4.11 Chi Squared Goodness of Fit and the T-Test</a></label></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> 4.8 Binomial Distribution <a href="#" class="mark-page-favorite pull-right" data-pid="2403" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsapplications.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../1065/statistics-and-probability.html">Statistics and Probability</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">4.8 Binomial Distribution</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <div class="intro-card" readonly="false"><img class="intro-image" readonly="true" src="../../factory-1.png" style="width: 460px; height: 331px;"> <div class="content" readonly="true">In this chapter we are going to explore probabilities in events where there are only two possible outcomes. This could be success or failure, truth or lies, accuracy or inaccuracy, etc. These particular types of events share characteristics that mean we can calculate probability using the binomial distribution.</div> </div> <div class="panel panel-has-colored-body panel-turquoise"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <p><span id="docs-internal-guid-486de6ee-7fff-a9d1-4896-02e9367973d0"></span></p> <p dir="ltr"><span id="docs-internal-guid-486de6ee-7fff-a9d1-4896-02e9367973d0">In this unit you should learn to…</span></p> <ul style="list-style-type:disc;"> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">understand what types of data are binomially distributed</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">be familiar using binomial notation</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">calculate the expected value, standard deviation and variance of a binomially distributed event</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">calculate probability using the binomial distribution</p> </li> <li aria-level="1" dir="ltr"> <p dir="ltr" role="presentation">calculate probability using the binomial distribution</p> </li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-yellow"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>1. Introduction</p> </div> </div> <div class="panel-body"> <div> <p>What types of data are binomially distributed? Notation and examples.</p> <p style="text-align: center"> </p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/527407187?badge=0&autopause=0&player_id=0&app_id=58479&h=f1623e1a8b"></iframe></div> <script src="../../../api/player.js"></script> <p> </p> <p> </p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>2. Expected Value, variance and standard deviation</p> </div> </div> <div class="panel-body"> <p>How can we use our knowledge to calculate the expected value, variance and standard deviation of an event?</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p> </p> </section> <p style="text-align: center"> </p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/598381435?badge=0&autopause=0&player_id=0&app_id=58479&h=d0b4594472"></iframe></div> <script src="../../../api/player.js"></script> <p> </p> <p> </p> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Calculating Probability using the binomial distribution</p> </div> </div> <div class="panel-body"> <div> <p>How can we calculate the probability of a particular number of outcomes occurring?</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p style="text-align: center"> </p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/598399283?badge=0&autopause=0&player_id=0&app_id=58479&h=1245fff605"></iframe></div> <script src="../../../api/player.js"></script> <p> </p> <p> </p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-yellow panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>3. Calculating Probability using the binomial distribution</p> </div> </div> <div class="panel-body"> <div> <p>How can we calculate the probability of a certain range of numbers of outcomes occurring?</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p style="text-align: center"> </p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/598411534?badge=0&autopause=0&player_id=0&app_id=58479&h=d96c45cbbf"></iframe></div> <script src="../../../api/player.js"></script> <p> </p> </section> </div> </div> </div> </div> </div> <h4> </h4> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <p> </p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div id="carousel-266" class="dynamic-gallery carousel slide" data-id="266"><div class="carousel-inner" role="listbox"><div class="item active"><a class="fancy" href="../../../std-galleries/15-266/bd1.png" data-fancybox="gallery-266" title="" data-caption=""><img alt="" src="../../../std-galleries/15-266/bd1.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-266/bd2.png" data-fancybox="gallery-266" title="" data-caption=""><img alt="" src="../../../std-galleries/15-266/bd2.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-266/bd3.png" data-fancybox="gallery-266" title="" data-caption=""><img alt="" src="../../../std-galleries/15-266/bd3.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-266/bd4.png" data-fancybox="gallery-266" title="" data-caption=""><img alt="" src="../../../std-galleries/15-266/bd4.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-266/bd5.png" data-fancybox="gallery-266" title="" data-caption=""><img alt="" src="../../../std-galleries/15-266/bd5.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-266/bd7.png" data-fancybox="gallery-266" title="" data-caption=""><img alt="" src="../../../std-galleries/15-266/bd7.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-266/bd8.png" data-fancybox="gallery-266" title="" data-caption=""><img alt="" src="../../../std-galleries/15-266/bd8.png"></a></div><div class="item "><a class="fancy" href="../../../std-galleries/15-266/bd9.png" data-fancybox="gallery-266" title="" data-caption=""><img alt="" src="../../../std-galleries/15-266/bd9.png"></a></div></div><a class="left carousel-control" href="#carousel-266" role="button" data-slide="prev"><i class="fa fa-fw fa-chevron-left"></i></a><a class="right carousel-control" href="#carousel-266" role="button" data-slide="next"><i class="fa fa-fw fa-chevron-right"></i></a></div><ol class="std-carousel-indicators"><li data-index="0"><img title="Click to view" src="../../../std-galleries/15-266/bd1-thumb128.jpg"><li><li data-index="1"><img title="Click to view" src="../../../std-galleries/15-266/bd2-thumb128.jpg"><li><li data-index="2"><img title="Click to view" src="../../../std-galleries/15-266/bd3-thumb128.jpg"><li><li data-index="3"><img title="Click to view" src="../../../std-galleries/15-266/bd4-thumb128.jpg"><li><li data-index="4"><img title="Click to view" src="../../../std-galleries/15-266/bd5-thumb128.jpg"><li><li data-index="5"><img title="Click to view" src="../../../std-galleries/15-266/bd7-thumb128.jpg"><li><li data-index="6"><img title="Click to view" src="../../../std-galleries/15-266/bd8-thumb128.jpg"><li><li data-index="7"><img title="Click to view" src="../../../std-galleries/15-266/bd9-thumb128.jpg"><li></li></ol> <p> </p> </section> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test yourself</p> </div> </div> <div class="panel-body"> <p>Self Checking Quiz</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <div class="tib-quiz" data-stats="15-524-2403"><div class="label label-default q-number">1</div><div class="exercise shadow-bottom"><div class="q-question"><p>For <span class="math-tex">\(X\sim B(n,p)\)</span>, describe what each element of this probability distribution represents. Use the following statements within the input boxes: discrete random variable, number of trials, probability of success, binomial distribution.</p></div><div class="q-answer"><p><span class="math-tex">\(X \)</span> is a <input type="text" style="height: auto;" data-c="discrete random variable"> <span class="review"></span></p><p><span class="math-tex">\(B\)</span> represents the <input type="text" style="height: auto;" data-c="binomial distribution"> <span class="review"></span></p><p><span class="math-tex">\(n\)</span> is the <input type="text" style="height: auto;" data-c="number of trials "> <span class="review"></span></p><p><span class="math-tex">\(p\)</span> is the <input type="text" style="height: auto;" data-c="probability of success"> <span class="review"></span></p></div><div class="q-explanation"><p>This is clear from the answers above.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">2</div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(X\sim B(10,0.3)\)</span>, find <span class="math-tex">\(P(X=4)\)</span>. Give your answer to 1 decimal place.</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="0.2"> <span class="review"></span> </p></div><div class="q-explanation"><p>Using the binomial distribution (pdf) on your GDC, you instantly arrive at the answer 0.2.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">3</div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(X\sim B(10,0.3)\)</span>, find <span class="math-tex">\(P\left(X\le3\right)\)</span>. Give your answer to 3 significant figures.</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 0.650</label></p><p><label class="radio"><input type="radio"> 0.382</label></p><p><label class="radio"><input type="radio"> 0.649</label></p><p><label class="radio"><input type="radio"> 0.383</label></p></div><div class="q-explanation"><p>Using the binomial distribution (cdf) on your GDC, ensuring that you input "3" into your calculator as this value must be included, you instantly arrive at the answer 0.6496107..., which is 0.650 written to 3 significant figures.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">4</div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(X\sim B(8,0.25)\)</span>, find <span class="math-tex">\(P\left(X<5\right)\)</span>. Give your answer to 3 significant figures.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 0.996</label></p><p><label class="radio"><input type="radio"> 0.995</label></p><p><label class="radio"><input type="radio"> 0.972</label></p><p><label class="radio"><input class="c" type="radio"> 0.973</label></p></div><div class="q-explanation"><p>Using the binomial distribution (cdf) on your GDC, ensuring that you input "4" into your calculator given that "5" is not included, you instantly arrive at the answer 0.97270..., which is 0.973 written to 3 significant figures.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">5</div><div class="exercise shadow-bottom"><div class="q-question"><p>There are 20 students in a class. Assuming that being born on any day of the week is equally likely, find the probability that exactly 3 of the students were born on a Monday. Give your answer to 3 significant figures.</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="0.242"> <span class="review"></span></p></div><div class="q-explanation"><p>This is a binomial distribution, with <span class="math-tex">\(X\)</span> representing the number of students born on a Monday. Hence, <span class="math-tex">\(X\sim B(20,\frac { 1 }{ 7 } )\)</span>, since there are 20 students and the probability of being born on a Monday is <span class="math-tex">\(\frac { 1 }{ 7 } \)</span>. The question is then, find <span class="math-tex">\(P(X=3)\)</span>, which using binomial (pdf) on the GDC, gives 0.242 to 3 significant figures.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">6</div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(X\sim B(20,0.1)\)</span>. Identify which sum below is equivalent to <span class="math-tex">\(P\left(X>18\right)\)</span>.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(P\left(X>18\right)=P(X=19)+P(X=20)+P(X=21)\)</span></label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(P\left(X>18\right)=P(X=19)+P(X=20)\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(P\left(X>18\right)=P(X=18)+P(X=19)+P(X=20)\)</span></label></p></div><div class="q-explanation"><p>See answer given above.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">7</div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(X\sim B(8,0.25)\)</span>, find <span class="math-tex">\(P\left(X=5\right)\)</span>. Give your answer to 3 significant figures.</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> 0.231</label></p><p><label class="radio"><input type="radio"> 0.973</label></p><p><label class="radio"><input type="radio"> 0.230</label></p><p><label class="radio"><input type="radio"> 0.996</label></p></div><div class="q-explanation"><p>Use the binomial distribution (pdf) on your GDC, and make sure to round carefully to 3sf</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">8</div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(X\sim B(9,0.3)\)</span>, find <span class="math-tex">\(P\left(X\le3\right)\)</span>. Give your answer to 3 significant figures.</p></div><div class="q-answer"><p><label class="radio"><input type="radio"> 0.729</label></p><p><label class="radio"><input type="radio"> 0.270</label></p><p><label class="radio"><input type="radio"> 0.463</label></p><p><label class="radio"><input class="c" type="radio"> 0.730</label></p></div><div class="q-explanation"><p>Using the binomial distribution (cdf) on your GDC, ensuring that you input "3" into your calculator as this value must be included, you instantly arrive at the answer 0.7296..., which is 0.730 written to 3 significant figures.</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="label label-default q-number">9</div><div class="exercise shadow-bottom"><div class="q-question"><p>Given that <span class="math-tex">\(X\sim B(10,0.3)\)</span>, find <span class="math-tex">\(P(X=3)\)</span>. Give your answer to 3sf</p></div><div class="q-answer"><p><input type="text" style="height: auto;" data-c="0.267"> <span class="review"></span></p></div><div class="q-explanation"><p>Using the binomial distribution (pdf) on your GDC, you instantly arrive at the answer 0.267. See calculator skills page for more help with this!</p></div><div class="actions"><span class="score" data-score="0"></span><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="totals"><span class="score"></span><button class="btn btn-success btn-block text-center check-total"><i class="fa fa-check-square-o"></i> Check</button></div></div><hr> </section> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam Style Questions</p> </div> </div> <div class="panel-body"> <p>The following questions are based on IB exam style questions from past exams. You should print these off (from the document at the top) and try to do these questions under exam conditions. Then you can check your work with the video solution.</p> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <p dir="ltr"><span id="docs-internal-guid-36defaf0-7fff-8511-6682-d0390141e64c">Ms Roberts teaches a Maths class with 17 students. 6 of them are working from school, and 11 of them are working remotely. </span></p> <p dir="ltr"><span id="docs-internal-guid-36defaf0-7fff-8511-6682-d0390141e64c">Each day, Ms Roberts chooses one student at random to make a presentation. </span></p> <p dir="ltr" role="presentation"><span id="docs-internal-guid-36defaf0-7fff-8511-6682-d0390141e64c">a. Find the probability that on a given day, Ms Roberts chooses someone who is working from home </span> <span id="docs-internal-guid-36defaf0-7fff-8511-6682-d0390141e64c">(1)</span></p> <p dir="ltr"><span id="docs-internal-guid-36defaf0-7fff-8511-6682-d0390141e64c">In September, Ms Roberts will teach 15 lessons with her class. </span></p> <p dir="ltr" role="presentation"><span id="docs-internal-guid-36defaf0-7fff-8511-6682-d0390141e64c">b. Find the probability that she chooses a student working remotely 8 times. </span><span id="docs-internal-guid-36defaf0-7fff-8511-6682-d0390141e64c">(2)</span><span id="docs-internal-guid-36defaf0-7fff-8511-6682-d0390141e64c"></span></p> <p dir="ltr" role="presentation"><span>c. Find the probability that she chooses a student working in school at least 9 times. </span><span id="docs-internal-guid-36defaf0-7fff-8511-6682-d0390141e64c">(3)</span></p> <p dir="ltr" style="margin-left: 36pt; text-align: right;">(6 marks)</p> <p dir="ltr" style="margin-left: 72pt;">Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p style="text-align: center"> </p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/598418978?badge=0&autopause=0&player_id=0&app_id=58479&h=f6ebe39357"></iframe></div> <script src="../../../api/player.js"></script> <script src="../../../api/player.js"></script></section> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <p dir="ltr"><span id="docs-internal-guid-a588dda0-7fff-53f6-5a7c-fe1dd27ff1ea">The probability that my train is late on any given day is 0.3. What is the probability that it will be late at least three times between Monday and Friday? </span><span id="docs-internal-guid-a588dda0-7fff-53f6-5a7c-fe1dd27ff1ea">(4)</span></p> <p dir="ltr">Video Solution</p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p style="text-align: center"> </p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/598421188?badge=0&autopause=0&player_id=0&app_id=58479&h=a05f2e4691"></iframe></div> <script src="../../../api/player.js"></script> <script src="../../../api/player.js"></script></section> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <p dir="ltr"><span id="docs-internal-guid-17bf6d9e-7fff-d0f6-7197-d2ebd3d87ae3">An infectious virus is spreading through a school. The probability of a randomly selected student having the virus next week is 0.2. Mr Santos has a class of 24 students. </span></p> <p dir="ltr" role="presentation"><span id="docs-internal-guid-17bf6d9e-7fff-d0f6-7197-d2ebd3d87ae3">a. Calculate the probability that 3 or more students in Mr Santos’ class will have the virus next week. </span> <span id="docs-internal-guid-17bf6d9e-7fff-d0f6-7197-d2ebd3d87ae3">(3)</span></p> <p dir="ltr" role="presentation"><span id="docs-internal-guid-17bf6d9e-7fff-d0f6-7197-d2ebd3d87ae3">b. If more than 25% of the students have the virus next week, the exam will be cancelled. What is the probability that the exam is cancelled? </span> <span id="docs-internal-guid-17bf6d9e-7fff-d0f6-7197-d2ebd3d87ae3">(4)</span></p> <p style="text-align: right;"><span id="docs-internal-guid-17bf6d9e-7fff-d0f6-7197-d2ebd3d87ae3">(6 marks)</span></p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p style="text-align: center"> </p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/598423906?badge=0&autopause=0&player_id=0&app_id=58479&h=7a7ef8304f"></iframe></div> <script src="../../../api/player.js"></script> <script src="../../../api/player.js"></script></section> </div> <div class="panel panel-has-colored-body panel-has-border panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <p dir="ltr"><span id="docs-internal-guid-2c708443-7fff-8c03-8aa8-0f628ddeade1">A wind turbine has 15 different working components, and works as long as at least of them is working. The probability that an individual component fails in a given year is 0.9. Given that the probability of each individual component failing is independent of the others:</span></p> <p dir="ltr" role="presentation"><span id="docs-internal-guid-2c708443-7fff-8c03-8aa8-0f628ddeade1">Find the probability that the wind turbine will fail in the first year. </span><span id="docs-internal-guid-2c708443-7fff-8c03-8aa8-0f628ddeade1">(2)</span></p> <p dir="ltr" role="presentation"> </p> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p style="text-align: center"> </p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/598427212?badge=0&autopause=0&player_id=0&app_id=58479&h=2853e96e30"></iframe></div> <script src="../../../api/player.js"></script> <p> </p> </section> </div> </div> <div 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