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<p><img alt="" src="../../files/probability/conditional-probability/main-1.jpg" style="float: left; width: 100px; height: 100px;"></p> <p>You may well have tried the Monty Hall problem, a question based on the game show 'Let's Make a Deal'. In the game show, extra information is given about the whereabouts of a prize. If you use this extra information, you are more likely to choose the correct strategy and win the car! This is the basis of conditional probability: extra information changes probabilities. You should be able to recognise these sort of questions, as they often contain the words "given that".</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <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>On this page, you should learn about</p> <ul> <li>conditional probability</li> <li>using Venn diagrams to visualise conditional probability, the probability of A given B</li> </ul> <p style="margin-left: 120px;"><img alt="" src="../../files/probability/conditional-probability/venn.jpg" style="width: 350px; height: 214px;"></p> <p style="margin-left: 120px;"><span class="math-tex">\(P(A|B)=\frac { P(A\cap B) }{ P(B) } \)</span></p> <p style="margin-left: 120px;"><span class="math-tex">\(P(A|B)=\frac { c }{ c+b } \)</span></p> <ul> <li>Bayes theorem</li> </ul> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/probability/conditional-probability/revision-notes_conditional_probability.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/probability/conditional-probability/revision-notes_conditional_probability.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>You can practise some of the skills in this page with the following quiz</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#98bd11bd"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="98bd11bd"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Conditional Probability <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-223-689" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Probability that the outcome of rolling a die is</p><p> <span class="q-text-draggable draggable" draggable="true">odd and a prime number<a aria-labelledby="cke_499_label" href="javascript:void(0)" id="cke_500_uiElement" role="button" style="-moz-user-select: none;" title="OK"><span id="cke_499_label"></span></a></span> <span class="q-text-draggable draggable" draggable="true">odd or a prime number</span> <span class="q-text-draggable draggable" draggable="true">odd if it is a prime number</span> </p><p>Put the statement next to the correct mathematical notation</p></div><div class="q-answer"><p><span class="math-tex">\(P(A\cap B)\)</span> = <input type="text" style="height: auto;" data-c="odd and a prime number"> <span class="review"></span> </p><p><span class="math-tex">\(P(A|B)\)</span> = <input type="text" style="height: auto;" data-c="odd if it is a prime number"> <span class="review"></span> </p><p><span class="math-tex">\(P(A\cup B)\)</span> = <input type="text" style="height: auto;" data-c="odd or a prime number"> <span class="review"></span> </p></div><div class="q-explanation"></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The following two-way table shows the number of females and males in grade 11 and 12 of a school</p><p>Complete the table and answer the probability questions below. The answers to the probabilities have been simplified.</p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: center;"> </td><td style="text-align: center;">Grade 11<hr class="hidden-separator"></td><td style="text-align: center;">Grade 12<hr class="hidden-separator"></td><td style="text-align: center;">Total<hr class="hidden-separator"></td></tr><tr><td style="text-align: center;">Females</td><td style="text-align: center;">15<hr class="hidden-separator"></td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="8"> <span class="review"></span><hr class="hidden-separator"></td><td style="text-align: center;">23<hr class="hidden-separator"></td></tr><tr><td style="text-align: center;">Males</td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="16"> <span class="review"></span><hr class="hidden-separator"></td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="16"> <span class="review"></span><hr class="hidden-separator"></td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="32"> <span class="review"></span><hr class="hidden-separator"></td></tr><tr><td style="text-align: center;">Total</td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="31"> <span class="review"></span></td><td style="text-align: center;">24</td><td style="text-align: center;">55</td></tr></tbody></table><p>A student is selected at random to represent the student group on the school council. The probability that the student is</p><p>a) female and from grade 12 = <span class="math-tex">\(\frac{a}{b}\)</span></p><p>a =<input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="55"> <span class="review"></span></p><p>b) male given that he/she is from grade 11 = <span class="math-tex">\(\frac{c}{d}\)</span></p><p>c =<input type="text" style="height: auto;" data-c="16"> <span class="review"></span></p><p>d = <input type="text" style="height: auto;" data-c="31"> <span class="review"></span></p><p>c) a grade 11 student given that he is male = <span class="math-tex">\(\frac{e}{f}\)</span></p><p>e =<input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p>f = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The possibility space below shows the outcomes when two dice are rolled and their scores are added together</p><p><img alt="" src="../../files/probability/conditional-probability/quiz/q3.jpg" style="width: 350px; height: 194px;"></p><p>The answers to the probabilities below have been simplified.</p></div><div class="q-answer"><p>Probability that the sum equals 5 = <span class="math-tex">\(\frac{a}{b}\)</span></p><p>a =<input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p><hr class="hidden-separator"><p>Probability that the sum is less than 6 =<span class="math-tex">\(\frac{c}{d}\)</span></p><p>c =<input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><p>d = <input type="text" style="height: auto;" data-c="18"> <span class="review"></span></p><hr class="hidden-separator"><p>Probability that the sum equals 5 given that the sum is less than 6 =<span class="math-tex">\(\frac{e}{f}\)</span></p><p>e =<input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p>f = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/probability/conditional-probability/quiz/q3a.jpg" style="width: 250px; height: 139px;"></p><p><img alt="" src="../../files/probability/conditional-probability/quiz/q3b.jpg" style="width: 250px; height: 139px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>The possibility space below shows the outcomes when two dice are rolled and their scores are multiplied together</p><p><img alt="" src="../../files/probability/conditional-probability/quiz/q4.jpg" style="width: 350px; height: 194px;"></p><p>The answers to the probabilities below have been simplified.</p></div><div class="q-answer"><p>Probability that the product is a multiple of 4 = <span class="math-tex">\(\frac{a}{b}\)</span></p><p>a =<input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="12"> <span class="review"></span></p><hr class="hidden-separator"><p>Probability that the product is a multiple of 3 =<span class="math-tex">\(\frac{c}{d}\)</span></p><p>c =<input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><p>d = <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p><hr class="hidden-separator"><p>Probability that the product is a multiple of 3 and the product is a multiple of 4 =<span class="math-tex">\(\frac{e}{f}\)</span></p><p>e =<input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p><p>f = <input type="text" style="height: auto;" data-c="36"> <span class="review"></span></p><hr class="hidden-separator"><p>Probability that the product is a multiple of 3 given that the product is a multiple of 4 =<span class="math-tex">\(\frac{g}{h}\)</span></p><p>g =<input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p><p>h = <input type="text" style="height: auto;" data-c="15"> <span class="review"></span></p></div><div class="q-explanation"><p>a)</p><p><img alt="" src="../../files/probability/conditional-probability/quiz/q4a.jpg" style="width: 250px; height: 139px;"></p><p>b)</p><p><img alt="" src="../../files/probability/conditional-probability/quiz/q4b.jpg" style="width: 250px; height: 139px;"></p><p>c)</p><p><img alt="" src="../../files/probability/conditional-probability/quiz/q4c1.jpg" style="width: 250px; height: 138px;"></p><p>d)</p><p><img alt="" src="../../files/probability/conditional-probability/quiz/q4c.jpg" style="width: 250px; height: 138px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>In a class of 50 students, 21 study Visual Arts (A), 27 study Business Studies (B) and 12 study neither.</p><p><img alt="" src="../../files/probability/conditional-probability/quiz/venn-diagram.jpg" style="width: 300px; height: 183px;"></p></div><div class="q-answer"><p>a) Work out the values of a , b , c and d in the Venn diagram</p><p>a = <input type="text" style="height: auto;" data-c="11"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="17"> <span class="review"></span></p><p>c = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span></p><p>d = <input type="text" style="height: auto;" data-c="12"> <span class="review"></span></p><p>b) Find the answers below if the fractions have been simplified.</p><p>If a student is selected at random, the probability that she/he...</p><ul><li>studies Visual Arts <strong>AND</strong> Business Studies = <span class="math-tex">\(\frac{e}{f}\)</span> </li></ul><p>e = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span> f = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><hr class="hidden-separator"><ul><li>studies Visual Arts <strong>given that </strong>she/he studies Business Studies =<span class="math-tex">\(\frac{g}{h}\)</span></li></ul><p>g = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span> h = <input type="text" style="height: auto;" data-c="27"> <span class="review"></span></p><hr class="hidden-separator"><ul><li>studies Business Studies <strong>given that </strong>she/he studies Visual Arts =<span class="math-tex">\(\frac{i}{j}\)</span></li></ul><p>i = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span> j = <input type="text" style="height: auto;" data-c="21"> <span class="review"></span></p></div><div class="q-explanation"><p>a)</p><p>There are 50 students altogether and 12 that take neither Visual Arts nor Business Studies. Hence there are 38 (50 - 12) who take either of the two subjects. Within the 21 students that study Visual Arts there are some that study Business Studies. We find the number in the intersection, c</p><p>21 + 27 - c = 38</p><p>b)</p><p>studies Visual Arts <strong>AND</strong> Business Studies = <span class="math-tex">\(\frac{10}{50}=\frac{1}{5}\)</span> </p><p>studies Visual Arts <strong>given that </strong>she/he studies Business Studies =<span class="math-tex">\(P(A|B)=\frac { P(A\cap B) }{ P(B) } =\frac{\frac{10}{50}}{\frac{27}{50}}=\frac{10}{27}\)</span></p><p>studies Business Studies <strong>given that </strong>she/he studies Visual Arts =<span class="math-tex">\(P(B|A)=\frac { P(B\cap A) }{ P(A) } =\frac{\frac{10}{50}}{\frac{21}{50}}=\frac{10}{21}\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>In a class of 24 students 16 students own a games console and 14 students own a scooter, 3 own neither.</p></div><div class="q-answer"><p>The probability that a students owns both a games console and a scooter = <span class="math-tex">\(\frac{a}{b}\)</span></p><p>a =<input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p><hr class="hidden-separator"><p>The probability that a students owns a games console given she/he owns a scooter = <span class="math-tex">\(\frac{c}{d}\)</span></p><p>c =<input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p><p>d = <input type="text" style="height: auto;" data-c="14"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/probability/conditional-probability/quiz/q6.jpg" style="width: 250px; height: 148px;"></p><p>a) <span class="math-tex">\(\frac{9}{24}\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>For the events A and B, it is known that P(A) = 0.6, P(B) = 0.5 and P(A') = 0.4</p><p>Give the following answers as decimals</p></div><div class="q-answer"><p>P(A<span class="math-tex">\(\cap \)</span>B) = <input type="text" style="height: auto;" data-c="0.2"> <span class="review"></span></p><p>P(A|B) = <input type="text" style="height: auto;" data-c="0.4"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/probability/conditional-probability/quiz/q7.jpg" style="width: 250px; height: 149px;"></p><p><span class="math-tex">\(P(A|B)=\frac { P(A\cap B) }{ P(B) } =\frac{0.2}{0.5}=\frac{2}{5}=0.4\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>A number is selected at random from the list below</p><p>1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10</p><p>Giving your answer as a decimal, find the probability that the number selected is</p></div><div class="q-answer"><p>a) less than 3 given that it is less than 6 = <input type="text" style="height: auto;" data-c="0.4"> <span class="review"></span></p><p>b) less than 6 given that it is more than 2 = <input type="text" style="height: auto;" data-c="0.375"> <span class="review"></span></p><p>c) less than 3 given that it is even = <input type="text" style="height: auto;" data-c="0.2"> <span class="review"></span></p><p>d) prime given that it is odd = <input type="text" style="height: auto;" data-c="0.6"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/probability/conditional-probability/quiz/q8.jpg" style="width: 250px; height: 153px;"></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>For the events A and B, it is known that P(A) = 0.45, P(B) = 0.5 and P(A<span class="math-tex">\(\cap\)</span>B)<strong>'</strong> = 0.8</p><p>The following answers as fractions in their simplest form</p></div><div class="q-answer"><p>a) P(A<span class="math-tex">\(\cap\)</span>B) = <span class="math-tex">\(\frac{a}{b}\)</span></p><p>a =<input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><hr class="hidden-separator"><p>b) P(B|A) = <span class="math-tex">\(\frac{c}{d}\)</span></p><p>c = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><p>d = <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p><hr class="hidden-separator"><p>c) P(A'|B') = <span class="math-tex">\(\frac{e}{f}\)</span></p><p>e = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p>f = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><hr class="hidden-separator"></div><div class="q-explanation"><p><img alt="" src="../../files/probability/conditional-probability/quiz/q9.jpg" style="width: 250px; height: 148px;"></p><p>c) P(A'|B') = <span class="math-tex">\(\frac { P(A'\cap B') }{ P(B') } =\frac{0.25}{0.5}=\frac{1}{2}=0.5\)</span></p></div><div class="slide-q-actions"><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="exercise shadow-bottom"><div class="q-question"><p>You know that your mathematics teacher Mr O'Cop has 2 children. He tells you that he has a son called Rob.</p><p>Find the probability that Rob has a brother</p><p style="text-align: right;">Think about this carefully!</p></div><div class="q-answer"><p>The probability that Rob has a brother = <span class="math-tex">\(\frac{a}{b}\)</span></p><p>a = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p>b = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>You might have thought that the probability of having a brother is 0.5. Don't worry, most people give this wrong answer when asked this question!</p><p>Here are the possibilities for his 2 children.</p><p><img alt="" src="../../files/probability/conditional-probability/quiz/q10.jpg" style="width: 50px; height: 105px;"></p><p>If he has a son, then he cannot have 2 girls so there are 3 possiblities.</p><p>Only one would have Rob having a brother!</p></div><div class="slide-q-actions"><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> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </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"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <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"> <div> <div class="smart-object center" data-id="520"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Henri Tarr travels to school by bike <span class="math-tex">\(\frac{3}{5}\)</span> of the week and by car the rest of the time.</p> <p>If he travels by bike, the probability that he is late is <span class="math-tex">\(\frac{1}{8}\)</span>.</p> <p>If he travels by car, the probability that he is late is <span class="math-tex">\(\frac{3}{8}\)</span>.</p> <p>a) Copy and complete the following tree diagram</p> <p><img alt="" src="../../files/probability/conditional-probability/esq1.jpg" style="width: 507px; height: 357px;"></p> <p>b) Find the probability that Henri goes to school by car and is late for school.</p> <p>c) Find the probability that he is late for school.</p> <p>d) Given that he is late, find the probability that he travels to school by car.</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>c) Probability Late = P(car AND late) OR P(bike AND late)</p> <p><content>d) This is a question about conditional probability. Use the previous parts.</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/conditional-probability/esq1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/conditional-probability/esq1.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border 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"> <div> <div class="smart-object center" data-id="521"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A and B are independent events. P(A) = 0.3 and P(B) = 0.4</p> <p>a) Find <span class="math-tex">\(P(A'\cap B')\)</span></p> <p>b) Hence find P(A'|B')</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>A Venn diagram would be helpful.</p> <p>Since A and B are independent, <span class="math-tex">\(P(A\cap B)=P(A)\times P(B)\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/conditional-probability/esq2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/conditional-probability/esq2.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default 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>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="522"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <div> <p>In the Olympic Games, it is believed that 5% of sprinters are taking anabolic steroids. A test is developed that gives a positive result for an athlete who <strong><u>has </u></strong>taken anabolic steroids in the last month in 90% of cases. The test gives a positive result for an athlete who <strong><u>has not </u></strong>taken anabolic steroids in the last month in 15% of cases. A sprinter in chosen at random.</p> <p>a) Find the probability that she/he tests positive anabolic steroids</p> <p>b) Given that the athlete tests positive, find the probability that the athlete has taken anabolic steroids in the last month.</p> <hr class="hidden-separator"></div> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Put the information in a tree diagram</p> <p>This is Bayes' Thereom.</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/conditional-probability/esq3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/conditional-probability/esq3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <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> <div class="smart-object center" data-id="523"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>A and B are independent events.</p> <p>P(A) = P(B) = p</p> <p>Show that P(A'|B')=1-p</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>A and B are independent, therefore <span class="math-tex">\(P(A\cap B)=P(A)\times P(B)\)</span></p> <p><content>Put all the information in a Venn diagram</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/conditional-probability/esq4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/conditional-probability/esq4.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> <div 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