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Distribution</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"> Probability Calculations <a href="#" class="mark-page-favorite pull-right" data-pid="919" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../549/stats-probability.html">Stats & Probability</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Probability Calculations</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"> <p><img alt="" src="../../files/probability/probability_rules/main2-1.jpg" style="float: left; width: 100px; height: 100px;"> This page is aimed to help you with all the basics of probability. You will learn how to list outcomes, how to use Venn diagrams and probability trees. You will also find out about mutually exclusive events and independent events and get plenty of practice in the kind of probability questions that come up in the examinations.</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"> <div> <p>On this page, you should learn to</p> <ul> <li>the probability of an event</li> <li>complementary events</li> <li>expected number of occurrences</li> <li>sample spaces</li> <li>Venn diagrams</li> <li>tree diagrams</li> <li>combined events: OR, AND</li> <li>mutually exclusive events</li> <li>independent events</li> </ul> </div> </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/probability_rules/revision-notes_probability-calculations.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/probability/probability_rules/revision-notes_probability-calculations.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>Here is a quiz that gets you to practise set notation and Venn diagrams</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#55c2aeea"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="55c2aeea"> <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%;"> Venn Diagrams <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="11-363-919" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Describe the shaded region below using set notation</p><p><img alt="" src="../../files/probability/probability_rules/venn-quiz/q2a.png" style="width: 300px; height: 174px;"></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(A\cap B
\)</span></label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> A OR B</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(A\cup B\)</span></label></p><p><label class="radio"><input type="radio"> A|B</label></p></div><div class="q-explanation"><p>This is the intersection of A and B</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>Which of the following Venn diagrams represents <span class="math-tex">\(A\cup B\)</span></p><p>Select <strong><u>all</u></strong> the correct answers</p><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td><p style="text-align: center;"><strong>A</strong></p><p style="text-align: center;"><strong><img alt="" src="../../files/probability/probability_rules/venn-quiz/q2a.png" style="width: 300px; height: 174px;"></strong></p></td><td><p style="text-align: center;"><strong>B</strong></p><p style="text-align: center;"><strong><img alt="" src="../../files/probability/probability_rules/venn-quiz/q2b.png" style="width: 300px; height: 174px;"></strong></p></td></tr><tr><td><p style="text-align: center;"><strong>C</strong></p><p style="text-align: center;"><strong><img alt="" src="../../files/probability/probability_rules/venn-quiz/q2c.png" style="width: 300px; height: 174px;"></strong></p></td><td><p style="text-align: center;"><strong>D</strong></p><p style="text-align: center;"><strong><img alt="" src="../../files/probability/probability_rules/venn-quiz/q2d.png" style="width: 300px; height: 174px;"></strong></p></td></tr></tbody></table></div><div class="q-answer"><p><label class="checkbox" style=" float: left; margin-right: 40px; "><input class="c" type="checkbox"> B</label></p><p><label class="checkbox" style=" float: left; margin-right: 40px; "><input class="c" type="checkbox"> C</label></p><p><label class="checkbox" style=" float: left; margin-right: 40px; "><input class="c" type="checkbox"> D</label></p><p><label class="checkbox" style=" float: left; margin-right: 40px; "><input type="checkbox"> A</label></p></div><div class="q-explanation"><p><span class="math-tex">\(A\cup B\)</span> is the union of A and B. We shade what is in A and B.</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>Describe the shaded region below using set notation</p><p><img alt="" src="../../files/probability/probability_rules/venn-quiz/q3.png" style="width: 300px; height: 174px;"></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(A'\cap B\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(A'\cup B\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(A\cap B'\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\((A\cup B)\cap B\)</span></label></p></div><div class="q-explanation"><p>The answer is <span class="math-tex">\(A'\cap B\)</span></p><ul><li>Imagine shading A' and B</li><li>Which region is shaded both times</li></ul></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>Describe the shaded region below using set notation</p><p><img alt="" src="../../files/probability/probability_rules/venn-quiz/q4.png" style="width: 300px; height: 174px;"></p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(A\cup B'\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\((A\cup B)'\cap A\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(A'\cap B\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(A\cap B'\)</span></label></p></div><div class="q-explanation"><p>The answer is <span class="math-tex">\(A\cup B'\)</span></p><ul><li>Imagine shading A and B'</li><li>What region is shaded altogether.</li></ul></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>Describe the shaded region below using set notation</p><p>Select <strong><u>all</u></strong> the correct answers</p><p><img alt="" src="../../files/probability/probability_rules/venn-quiz/q5.png" style="width: 300px; height: 174px;"></p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\((A\cup B)'\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(A'\cup B'\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\((A\cap B)'\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(A'\cap B'\)</span></label></p></div><div class="q-explanation"><p>There are two correct answers that describe the same region.</p><p>For <span class="math-tex">\((A\cup B)'\)</span> , shade what is outside <span class="math-tex">\(A\cup B\)</span></p><p>For <span class="math-tex">\(A'\cap B'\)</span> , imagine shading A' and B'. The region is what has been shaded twice.</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>Describe the shaded region below using set notation</p><p>Select <strong><u>all</u></strong> the correct answers</p><p><img alt="" src="../../files/probability/probability_rules/venn-quiz/q6.png" style="width: 300px; height: 174px;"></p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\((A\cup B)'\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(A'\cup B'\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(A'\cap B'\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\((A\cap B)'\)</span></label></p></div><div class="q-explanation"><p>There are two correct answers that describe the same region</p><p>For <span class="math-tex">\((A\cap B)'\)</span> , shade what is outside <span class="math-tex">\(A\cap B\)</span></p><p>For <span class="math-tex">\(A'\cup B'\)</span>, shade A' and B'</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 is a subset of B. Describe the shaded region below using set notation</p><p>Select <strong><u>all</u></strong> the correct answers</p><p><img alt="" src="../../files/probability/probability_rules/venn-quiz/q7.png" style="width: 300px; height: 174px;"></p></div><div class="q-answer"><p><label class="checkbox" style=" float: left; margin-right: 40px; "><input class="c" type="checkbox"> A</label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(A\cap B'\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(A'\cap B\)</span></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(A\cap B\)</span></label></p></div><div class="q-explanation"><p>Since A is a subset of B, then <span class="math-tex">\(A\cap B\)</span> also represents this region</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 is a subset of B. Describe the shaded region below using set notation</p><p><img alt="" src="../../files/probability/probability_rules/venn-quiz/q8.png" style="width: 300px; height: 174px;"></p></div><div class="q-answer"><p><label class="radio"><input type="radio"> <span class="math-tex">\(A'\cup B\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(A\cap B'\)</span></label></p><p><label class="radio" style=" float: left; margin-right: 40px; "><input type="radio"> B</label></p><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(A'\cap B\)</span></label></p></div><div class="q-explanation"><p>This region is what is in B and not in A</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> <hr class="hidden-separator"> <p>Here is a quiz about mutually exclusive and independent events </p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#da4cfa98"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="da4cfa98"> <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%;"> Mutually Exclusive and Independent Events <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="11-357-919" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>For the two events A and B, <span class="math-tex">\(P(A \cap B)\)</span> = 0</p><p>Select <u><strong>all</strong></u> the answers that are true</p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> P(A) = 0 and/or P(B) = 0</label></p><p><label class="checkbox"><input type="checkbox"> A and B are independent events</label></p><p><label class="checkbox"><input class="c" type="checkbox"> A and B are mutually exclusive</label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(P(A \cup B) =P(A) + P(B)\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(P(A \cap B)\)</span> = 0 means that the two events are mutually exclusive</p><p>This means that <span class="math-tex">\(P(A \cup B) =P(A) + P(B) - P(A\cap B) \\ P(A \cup B) =P(A) + P(B) - 0\)</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 two events A and B, <span class="math-tex">\(P(A \cap B) = P(A) \times P(B)\)</span></p><p>Select <u><strong>all</strong></u> the answers that are true</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> P(A|B) = P(A)</label></p><p><label class="checkbox"><input class="c" type="checkbox"> A and B are independent events</label></p><p><label class="checkbox"><input type="checkbox"> A and B are mutually exclusive</label></p><p><label class="checkbox"><input type="checkbox"> P(A) + P(B) = <span class="math-tex">\(P(A\cup B)\)</span></label></p></div><div class="q-explanation"><p>If <span class="math-tex">\(P(A \cap B) = P(A) \times P(B)\)</span>, then A and B are independent</p><p>Since the probability of A happening is not affected by B, then P(A|B) = P(A)</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 below, select <u><strong>all</strong></u> the answers that are true</p><p><img alt="" src="../../files/probability/probability_rules/quiz/q3.png" style="width: 350px; height: 202px;"></p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> P(A) = 0 and/or P(B) = 0</label></p><p><label class="checkbox"><input class="c" type="checkbox"> A and B are mutually exclusive</label></p><p><label class="checkbox"><input type="checkbox"> A and B are independent events</label></p><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(P(A \cup B) =P(A) + P(B)\)</span></label></p></div><div class="q-explanation"><p>The two events are mutually exclusive and <span class="math-tex">\(P(A \cap B)\)</span>= 0</p><p>This means that <span class="math-tex">\(P(A \cup B) =P(A) + P(B) - P(A\cap B) \\ P(A \cup B) =P(A) + P(B) - 0\)</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>The two events A and B are mutually exclusive. Find the value of <strong><em>x</em></strong></p><p><strong><em><img alt="" src="../../files/probability/probability_rules/quiz/q4.png" style="width: 350px; height: 202px;"></em></strong></p></div><div class="q-answer"><p><strong><em>x</em></strong> = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p>The two events are mutually exclusive and therefore <span class="math-tex">\(P(A \cap B)\)</span> = 0</p><p>The Venn diagram could be drawn with no intersection!</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 and B are independent events.</p><p>P(A) = 0.3 , P(B') = 0.6.</p><p>Work out <span class="math-tex">\(P(A \cap B)\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(P(A \cap B)\)</span> = <input type="text" style="height: auto;" data-c="0.12"> <span class="review"></span></p></div><div class="q-explanation"><p>P(B) = 1 - 0.6 = 0.4</p><p><span class="math-tex">\(P(A \cap B) = P(A) \times P(B)\)</span> = 0.3 x 0.4 = 0.12</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 and B are independent events.</p><p>P(A) = 0.6 , <span class="math-tex">\(P(A \cap B)\)</span> = 0.54</p><p>Work out P(B)</p></div><div class="q-answer"><p>P(B) = <input type="text" style="height: auto;" data-c="0.9"> <span class="review"></span> </p></div><div class="q-explanation"><p><span class="math-tex">\(P(A \cap B) = P(A) \times P(B)\)</span></p><p>P(B) = 0.54 <span class="math-tex">\(\div \)</span> 0.6 = 0.9</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 and B are independent events.</p><p>P(A') = 0.6 , P(B) = 0.6</p><p>Work out <span class="math-tex">\(P(A \cup B) \)</span></p></div><div class="q-answer"><p><span class="math-tex">\(P(A \cup B) \)</span> = <input type="text" style="height: auto;" data-c="0.76"> <span class="review"></span> </p></div><div class="q-explanation"><p>P(A) = 0.4</p><p><span class="math-tex">\(P(A \cap B) = P(A) \times P(B)\)</span> = 0.4x0.6 = 0.24</p><p><span class="math-tex">\(P(A \cup B) =P(A) + P(B) - P(A\cap B) \)</span></p><p><span class="math-tex">\(P(A \cup B) \)</span> = 0.4 + 0.6 - 0.24 = 0.76</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 and B are mutually exclusive events</p><p>P(A')= 0.6 , P(B') = 0.65</p><p>Work out <span class="math-tex">\(P(A\cup B)\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(P(A\cup B)\)</span> = <input type="text" style="height: auto;" data-c="0.75"> <span class="review"></span> </p></div><div class="q-explanation"><p>P(A) = 0.4</p><p>P(B) = 0.35</p><p><span class="math-tex">\(P(A\cup B)\)</span> = 0.4 + 0.35</p><hr class="hidden-separator">Note that A' and B' are <strong><u>not</u></strong> mutually exclusive</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">A and B are independent events<p>P(A')= 0.5 and P(B') = 0.4</p><p>Work out <span class="math-tex">\(P(A' \cap B')\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(P(A' \cap B')\)</span> = <input type="text" style="height: auto;" data-c="0.2"> <span class="review"></span></p></div><div class="q-explanation"><p>P(A) = 0.5 , P(B) = 0.6</p><p><span class="math-tex">\(P(A\cap B)\)</span> = 0.5x0.6 = 0.3</p><p><img alt="" src="../../files/probability/probability_rules/quiz/q9.png" style="width: 300px; height: 176px;"></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 and B are independent events</p><p>P(A) = 0.3 and P(B) = 0.6</p><p>Find P(A'|B)</p></div><div class="q-answer"><p>P(A'|B) = <input type="text" style="height: auto;" data-c="0.7"> <span class="review"></span> </p></div><div class="q-explanation"><p>If A and B are independent events</p><p>P(A'|B) = P(A')</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> <hr class="hidden-separator"> <p>Here is a quiz about the probability rule <span class="math-tex">\(P(A\cup B)\)</span> = P(A) + P(B) - <span class="math-tex">\(P(A\cap B)\)</span> . You might find Venn diagrams helpful to answer these questions</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#88ae3358"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="88ae3358"> <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%;"> Probability Rules <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="11-366-919" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Consider two events A and B such that P(A) = 0.5 , P(B) = 0.6 , <span class="math-tex">\(P(A\cap B)\)</span> = 0.25.<br>Work out <span class="math-tex">\(P(A\cup B)\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(P(A\cup B)\)</span> = <input type="text" style="height: auto;" data-c="0.85"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(P(A\cup B)\)</span> = P(A) + P(B) - <span class="math-tex">\(P(A\cap B)\)</span></p><p>= 0.5 + 0.6 - 0.25 = 0.85</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>Consider two events A and B such that P(B) = 0.38 , <span class="math-tex">\(P(A\cup B)\)</span> = 0.89 , <span class="math-tex">\(P(A\cap B)\)</span> = 0.27.<br>Work out P(A)</p></div><div class="q-answer"><p>P(A) = <input type="text" style="height: auto;" data-c="0.78"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(P(A\cup B)\)</span> = P(A) + P(B) - <span class="math-tex">\(P(A\cap B)\)</span></p><p>0.89 = P(A) + 0.38 - 0.27</p><p>P(A) = 0.78</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>Consider two events A and B such that P(A) = 0.7 , P(B) = 0.2</p><p>Work out <span class="math-tex">\(P(A\cup B)\)</span> if A and B are independent events</p></div><div class="q-answer"><p><span class="math-tex">\(P(A\cup B)\)</span> = <input type="text" style="height: auto;" data-c="0.76"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(P(A\cap B)\)</span> = P(A)xP(B) = 0.7x0.2 = 0.14</p><p><span class="math-tex">\(P(A\cup B)\)</span> = P(A) + P(B) - <span class="math-tex">\(P(A\cap B)\)</span></p><p><span class="math-tex">\(P(A\cup B)\)</span> = 0.7 + 0.2 - 0.14 = 0.76</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>Consider two events A and B such that P(A) = 0.65 , P(B) = 0.31 , <span class="math-tex">\(P(A\cap B)\)</span> = 0.22</p><p>Work out <span class="math-tex">\(P(A'\cap B)\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(P(A'\cap B)\)</span> = <input type="text" style="height: auto;" data-c="0.09"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/probability/probability_rules/quiz3/q4.png" style="width: 300px; height: 174px;"></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>Consider two events A and B such that P(B) = 0.55 , <span class="math-tex">\(P(A\cap B)\)</span> = 0.22.</p><p>Work out <span class="math-tex">\(P(A\cap B')\)</span> if A and B are independent</p></div><div class="q-answer"><p><span class="math-tex">\(P(A\cap B')\)</span> = <input type="text" style="height: auto;" data-c="0.18"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(P(A\cap B)\)</span> = P(A)xP(B)</p><p>0.22 = P(A)x0.55</p><p>P(A) = 0.4</p><p><img alt="" src="../../files/probability/probability_rules/quiz3/q5a.png" style="width: 300px; height: 174px;"></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>Consider two events A and B such that P(A) = 0.7 , P(B) = 0.6</p><p>Work out <span class="math-tex">\(P(A'\cap B')\)</span> if A and B are independent events</p></div><div class="q-answer"><p><span class="math-tex">\(P(A'\cap B')\)</span> = <input type="text" style="height: auto;" data-c="0.12"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(P(A\cap B)\)</span> = 0.7x0.6 = 0.42</p><p><img alt="" src="../../files/probability/probability_rules/quiz3/q6.png" style="width: 300px; height: 174px;"></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>Consider two events A and B such that <span class="math-tex">\(P(A\cup B)'\)</span> = 0.22 , <span class="math-tex">\(P(A'\cap B)\)</span> = 0.34</p><p>Work out P(A)</p></div><div class="q-answer"><p>P(A) = <input type="text" style="height: auto;" data-c="0.44"> <span class="review"></span></p></div><div class="q-explanation"><p>P(A) = 1 - 0.34 - 0.22</p><p><img alt="" src="../../files/probability/probability_rules/quiz3/q7.png" style="width: 300px; height: 174px;"></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>Consider two events A and B such that P(A) = 0.5 , P(B) = 0.7.</p><p>Work out <span class="math-tex">\(P(A'\cup B)\)</span> if A and B are independent events</p></div><div class="q-answer"><p><span class="math-tex">\(P(A\cap B)\)</span> = <input type="text" style="height: auto;" data-c="0.85"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(P(A\cap B)\)</span> = 0.5 x 0.7 = 0.35</p><p><img alt="" src="../../files/probability/probability_rules/quiz3/q8a.png" style="width: 300px; height: 174px;"></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>Consider two events A and B such that P(A) = 0.3 , P(B) = 0.5.</p><p>Find <span class="math-tex">\(P(A\cup B)\)</span> if <span class="math-tex">\(P(A\cup B) = 3P(A\cap B)\)</span></p></div><div class="q-answer"><p><span class="math-tex">\(P(A\cup B)\)</span> = <input type="text" style="height: auto;" data-c="0.6"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(P(A\cup B)\)</span> = P(A) + P(B) - <span class="math-tex">\(P(A\cap B)\)</span></p><p><span class="math-tex">\( 3P(A\cap B)\)</span> = 0.3 + 0.5 - <span class="math-tex">\( P(A\cap B)\)</span></p><p>Treat this as an equation and solve. You can even substitute <span class="math-tex">\(P(A\cap B)\)</span> = x</p><p>3x = 0.8 - x</p><p>4x = 0.8</p><p>x = 0.2</p><p><span class="math-tex">\(P(A\cap B)\)</span> = 0.2</p><p><span class="math-tex">\(P(A\cup B)\)</span> = 3 x 0.2 = 0.6</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>Consider two events A and B such that <span class="math-tex">\(P(A\cup B)\)</span> = 0.8 , P(A) = 0.6</p><p>Given that A and B are independent, find P(B)</p></div><div class="q-answer"><p>P(B) = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(P(A\cap B)\)</span> = 0.6P(B)</p><p><span class="math-tex">\(P(A\cup B)\)</span> = P(A) + P(B) - <span class="math-tex">\(P(A\cap B)\)</span></p><p>0.8 = 0.6 + P(B) - 0.6P(B)</p><p>Treat this as an equation and solve. You can even substitute P(B) = x</p><p>0.8 = 0.6 + x - 0.6x</p><p>0.2 = 0.4x</p><p>x = 0.5</p><p>P(B) = 0.5</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> <hr class="hidden-separator"> <p>The final quiz on this page is about independent events. You might find tree diagrams helpful to answer these questions</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#95b980e9"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="95b980e9"> <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%;"> Independent Events <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="11-369-919" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>A bag contains 3 red balls and 2 white balls. A ball is selected at random. Without replacing the first, a second ball is selected. Find the probability that 2 red balls are selected.</p></div><div class="q-answer"><p>probability = <input type="text" style="height: auto;" data-c="0.3"> <span class="review"></span></p></div><div class="q-explanation"><p>The probability that the first ball is red is <span class="math-tex">\(\frac {3}{5}\)</span></p><p>The probability that the second ball is red is <span class="math-tex">\(\frac {2}{4}\)</span></p><p>Probability of red AND red = <span class="math-tex">\(\frac {3}{5} \times \frac {2}{4} = \frac {6}{20}\)</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>Three fair coins are flipped. Find the probability that 2 heads are flipped.</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\frac {3}{8}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\frac {1}{8}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\frac {2}{3}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\frac {2}{8}\)</span></label></p></div><div class="q-explanation"><p>Here are all 8 possible outcomes</p><p>HHH</p><p><em><strong>HHT</strong></em></p><p><em><strong>HTH</strong></em></p><p><em><strong>THH</strong></em></p><p>TTH</p><p>THT</p><p>HTT</p><p>TTT</p><p>There are 3 outcomes with 2 heads</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 bag contains 3 green balls and 2 other coloured balls. A ball is taken at random then replaced. A second ball is taken. Find the probability that 2 green balls are taken.</p></div><div class="q-answer"><p>probability = <input type="text" style="height: auto;" data-c="0.36"> <span class="review"></span></p></div><div class="q-explanation"><p>probability = <span class="math-tex">\(\frac {3}{5} \times \frac {3}{5}=\frac {9}{25}\)</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 drawer contains 4 black socks and 6 stripey socks. Two socks are drawn at random. The probability that 2 black socks are taken is <span class="math-tex">\(\frac {a}{15}\)</span>.</p><p>Find <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a </em></strong>= <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\frac {4}{10}\times \frac {3}{9}=\frac {12}{90}=\frac {2}{15}\)</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 bag contains 7 black and 3 white balls. Two balls are taken at random.</p><p>The probability that both balls are the same colour is <span class="math-tex">\(\frac {a}{15}\)</span></p><p>Work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p>probability of BB OR WW = <span class="math-tex">\(\frac {7}{10}\times \frac {6}{9}+\frac {3}{10}\times \frac {2}{9}\)</span> = <span class="math-tex">\(\frac {42}{90}+\frac {6}{90}=\frac {48}{90}=\frac {8}{15}\)</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 drawer contains 2 black, 4 grey and 6 white socks. Two socks are drawn at random. Find the probability that they are the same colour.</p></div><div class="q-answer"><p><label class="checkbox"><input class="c" type="checkbox"> <span class="math-tex">\(\frac {44}{132}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\frac {56}{144}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\frac {44}{144}\)</span></label></p><p><label class="checkbox"><input type="checkbox"> <span class="math-tex">\(\frac {3}{12}\)</span></label></p></div><div class="q-explanation"><p>probability (BB OR GG OR WW) = <span class="math-tex">\(\frac {2}{12}\times\frac {1}{11}+\frac {4}{12}\times\frac {3}{11}+\frac {6}{12}\times\frac {5}{11}\)</span></p><p>= <span class="math-tex">\(\frac {2}{132}+\frac {12}{132}+\frac {30}{132}=\frac {44}{132}\)</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>The probability that a sportsperson is left-handed is 0.12.</p><p>The probability that a sportsperson is left-handed <strong>and</strong> left-footed is 0.102.</p><p>The probability that a sportsperson is right-handed <strong>and</strong> right-footed is 0.792.</p><p>Work out <strong><em>a</em></strong> , <strong><em>b</em></strong> , <strong><em>c </em></strong>and <strong><em>d </em></strong>in the probability tree below</p><p><img alt="" src="../../files/probability/probability_rules/independent-events/q7" style="width: 600px; height: 403px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="0.85"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="0.15"> <span class="review"></span></p><p><strong><em>c</em></strong> = <input type="text" style="height: auto;" data-c="0.1"> <span class="review"></span></p><p><strong><em>d</em></strong> = <input type="text" style="height: auto;" data-c="0.9"> <span class="review"></span></p></div><div class="q-explanation"><p><img alt="" src="../../files/probability/probability_rules/independent-events/q7a" style="width: 600px; height: 279px;"></p><p>0.12 x a = 0.102</p><p>a = 0.85</p><p>b = 0.15</p><hr class="hidden-separator"><p>0.88 x d = 0.792</p><p>d = 0.9</p><p>c = 0.1</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><img alt="" src="../../files/probability/probability_rules/independent-events/jars.png" style="width: 350px; height: 243px;"></p><p>Jar A contains 4 blue marbles and 2 red marbles.</p><p>Jar B contains 2 blue marbles and 2 red marbles.</p><p>A marble is taken at random from jar A and placed in jar B. Next, a marble is taken from jar B and placed in jar A.</p><p>After this, the probability that there are still 4 blue marbles in jar A is <span class="math-tex">\(\frac{a}{30}\)</span>.</p><p>Work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="18"> <span class="review"></span></p></div><div class="q-explanation"><p>There are two possibilities so that there are 4 marbles in jar A</p><p>1) A blue is taken from jar A and put into jar B (there will be 3 blue and 2 red in jar B)</p><p>2) A red is taken from jar A and put into jar B (there will be 2 blue and 3 red in jar B)</p><p>Probability (B1, B2 OR R1,R2) = <span class="math-tex">\(\frac{4}{6}\times \frac{3}{5}+\frac{2}{6}\times \frac{3}{5}\)</span> = <span class="math-tex">\(\frac{18}{30}\)</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>Maria drives to work.</p><p>The probability that she is late when it rains is 0.6</p><p>The probability that she is late when it does not rain is 0.1</p><p>In November, the probability that it rains on the drive to work is 0.4</p><hr class="hidden-separator"><p>a) On a particular day in November, find the probability that she is late for work</p><p>b) She will be disciplined at work if she is late (at least) twice in 3 consecutive days. It is a Monday morning in November, find the probability that she will disciplined on Wednesday.</p></div><div class="q-answer"><p>a) probability that she is late for work = <input type="text" style="height: auto;" data-c="0.3"> <span class="review"></span></p><p>b) probability that she will disciplined = <input type="text" style="height: auto;" data-c="0.216"> <span class="review"></span></p></div><div class="q-explanation"><p>a) probability (rain AND late OR not rain AND late) = 0.4 x 0.6 + 0.6 x 0.1 = 0.3</p><p>b) Let L = late , <span class="math-tex">\(\overline{L}\)</span> = not late</p><p>There are 4 possible outcomes to be disciplined</p><p><span class="math-tex">\(\overline{L}LL\\ L\overline{L}L\\ LL\overline{L}\\ LLL\)</span></p><p>Note that she will also be disciplined if she is late on all three days</p><p><span class="math-tex">\(3 \times0.3^2\times0.7 \ + \ 0.3^3\)</span> = 0.216</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 order to revise for her IB examinations, a student sits <strong><u>3</u></strong> practice papers.</p><p>The probability that she achieves a grade 7 on the first paper is 0.2</p><p>Whenever she gains a grade 7 in a paper, she becomes more confident and the probability of getting a grade 7 on the next paper <strong>increases by 0.1</strong></p><p>However, whenever she does not get a grade 7 in a paper, the probability of getting a grade 7 on the next paper <strong>decreases by 0.05</strong></p><p>Complete the following probability tree</p><p><img alt="" src="../../files/probability/probability_rules/independent-events/q10" style="width: 600px; height: 506px;"></p><p>Find the probability that she gets exactly two grade 7s on the practice papers.</p></div><div class="q-answer"><p>probability = <input type="text" style="height: auto;" data-c="0.101"> <span class="review"></span></p></div><div class="q-explanation"><p>Here is the completed probability tree showing the three possible ways of getting exactly 2 grade 7 results</p><p><img alt="" src="../../files/probability/probability_rules/independent-events/q10a.png" style="width: 700px; height: 401px;"></p><p>probability = 0.036 + 0.035 + 0.03 = 0.101</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="546"> <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-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <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> <p>Two events <strong><em>A</em></strong> and <strong><em>B </em></strong> are such that P(A) = 0.35 and P(B) = 0.6 and <span class="math-tex">\(P(A\cup B)\)</span> = 0.74</p> <p>a) Find <span class="math-tex">\(P(A\cap B)\)</span></p> <p>b) Determine whether A and B are independent</p> <p>c) Find <span class="math-tex">\(P(A'\cap B)\)</span></p> <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><content> </content>a) Put the information in a Venn diagram or use the rule <span class="math-tex">\(P(A\cup B)=P(A) +P(B)-P(A\cap B)\)</span></p> <p>b) If 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/probability_rules/esq_probability-rules1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/probability_rules/esq_probability-rules1.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="547"> <p><img alt="" height="222" src="../../files/probability/probability_rules/esq2.png" style="float: right;" width="108"><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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <div> <p>Katniss is practising archery. She fires three arrows at a target. The probability that she hits the target with her first arrow is 0.7. Whenever she hits the target, her confidence increases so that the probability that she hits the target on her next attempt increases by 0.1. Whenever she misses the target, the probability reduces by 0.1.</p> <p><img alt="" src="../../files/probability/probability_rules/esq2tree.jpg" style="width: 700px; height: 418px; float: left;">a) Complete the probability tree for Katniss’s three attempts.</p> <hr class="hidden-separator"></div> <div> <p>b) Calculate the probability that she hits the target with two attempts.</p> <p>c) Find the probability the she hits the target on at least one attempt.</p> </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>a) Fill out the probability tree carefully, the probability of hits and misses changes each time</p> <p><content>c) P(at least one hit) = 1 - P(zero hits)</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/probability_rules/esq_probability-rules2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/probability_rules/esq_probability-rules2.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="548"> <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-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>Events <strong><em>A</em></strong> and <strong><em>B</em></strong> are such that P(A) = 0.4 and <span class="math-tex">\(P(A\cup B)\)</span> = 0.7</p> <p>Find P(B) if <strong><em>A</em></strong> and <strong><em>B</em></strong> are independent</p> <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>If <strong><em>A</em></strong> and <strong><em>B</em></strong> are independent, then <span class="math-tex">\(P(A\cap B) =P(A)\times P(B)\)</span><content> </content></p> <p>Write an equation and consider writing P(B) = x</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/probability_rules/esq_probability-rules3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/probability_rules/esq_probability-rules3.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 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="549"> <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/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"></p> <div> <p>Alphonse and Bettina are playing a game. A bag contains 2 yellow beads and 3 red beads. They take it in turns to pick a bead from the bag at random. Alphonse goes first. If Alphonse picks a yellow bead, he wins and the game stops. If he picks a red bead, he replaces the bead and it is Bettina’s turn. If Bettina picks a red bead, she wins and the game stops. If she picks a yellow bead, she replaces the bead and it is Alphonse’s turn again.</p> <p style="text-align: center;"><img alt="" src="../../files/probability/probability_rules/esq4.jpg" style="width: 300px; height: 71px;"></p> <p>a) Find the probability that Alphonse wins on his first turn.</p> <p>b) Show that the probability that Alphonse wins on his second turn is <span class="math-tex">\(\frac{12}{125}\)</span></p> <p>c) The game continues until one of the player wins. What is the probability that Alphonse wins the game?</p> </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><content> </content>b) For Alphonse to win on his second turn, Alphonse must lose on his first turn, and then Bettina must lose on her first turn and then Alphonse must win on his second turn.</p> <p>c) The probability that Alphonse wins is the sum of an infinite geometric series <span class="math-tex">\({ S }_{ \infty }=\frac { { U }_{ 1 } }{ 1-r } \)</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/probability_rules/esq_probability-rules4.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/probability_rules/esq_probability-rules4.pdf" width="640"></iframe></p> </section> <h4> </h4> </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 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="550"> <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/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>A box contains 25 tickets. <strong><em>x</em></strong> tickets are gold, the rest are silver. <strong>Two </strong>tickets are selected at random.</p> <p>a) Show that the probability of selecting two gold tickets is <span class="math-tex">\(\frac{x^2-x}{600}\)</span></p> <p>b) Find the probability of selecting two tickets of the same colour.</p> <p>c) The probability of selecting two tickets of the same colour is twice the probability of selecting two tickets of a different colour. Find how many gold tickets there are.</p> </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><content> </content>a) There are 25 - <strong><em>x</em></strong> silver tickets</p> <p>b) Find probability of GG or SS</p> <p>c) If the probability of selecting two tickets of the same colour is twice the probability of selecting two tickets of a different colour, then the probability of selecting two tickets of the same colour = <span class="math-tex">\(\frac{2}{3}\)</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/probability_rules/esq_probability-rules5.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/probability_rules/esq_probability-rules5.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="919"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>Probability Calculations</strong> have you understood?</p><div class="slider-container text-center"><div id="self-assessment-slider" class="sib-slider self-assessment " data-value="1" data-percentage=""></div></div> <label class="label-lg">My notes</label> <textarea name="page-notes" class="form-control" rows="3" 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