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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"> Binomial Distribution <a href="#" class="mark-page-favorite pull-right" data-pid="692" 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">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"> <p><img alt="" src="../../files/probability/binomial-distribution/main-1.jpg" style="float: left; width: 100px; height: 100px;"></p> <p>The binomial distribution is an example of a discrete random variable. It has two parameters <strong>n </strong>(number of trials) and<strong> p </strong>(probability of success of one trial): <strong>X~B(n , p). </strong>For a situation to be described using a binomial model, the following must be true</p> <hr class="hidden-separator"> <ul> <li>A finite number of trials, n are carried out</li> <li>The trials are independent</li> <li>The outcome of each trial is deemed either a success or a failure</li> <li>The probability, p, of a successful outcome is the same for each trial</li> </ul> <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 about</p> <ul> <li>the binomial distribution</li> <li>mean of the binomial distribution</li> <li>variance of the binomial distribution</li> <li>the formula for calculating probabilities</li> </ul> <p style="margin-left: 120px;"><strong>P(X = r) = <sup>n</sup>C<sub>r</sub> x p<sup>r</sup> x (1 - p)<sup>n - r</sup></strong></p> </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/binomial-distribution/revision-notes_binomial_distribution.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/probability/binomial-distribution/revision-notes_binomial_distribution.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 practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#75d33160"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="75d33160"> <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%;"> Binomial Distribution <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="11-220-692" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>For a binomial distribution, some of the following statements are TRUE and some are FALSE.</p><p>Type T (TRUE) or F (FALSE) next to each one</p></div><div class="q-answer"><p>The probability of one trial depends on the outcome of the previous trial <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>A finite number of trials is carried out <input type="text" style="height: auto;" data-c="T"> <span class="review"></span></p><p>The distribution is always symmetrical <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>The binomial distribution is a continuous distribution <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>The outcome of each trial is deemed either a success or a failure <input type="text" style="height: auto;" data-c="T"> <span class="review"></span></p><p>The mean and variance of the distribution are always equal <input type="text" style="height: auto;" data-c="F"> <span class="review"></span></p><p>The probability of a successful outcome is the same for each trial <input type="text" style="height: auto;" data-c="T"> <span class="review"></span></p><p style="margin:0in;font-family:Tahoma;font-size:11.0pt">The trials are independent <input type="text" style="height: auto;" data-c="T"> <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 graph below shows a binomial distribution for the variable X. Answer the probability questions below</p><p><img alt="" src="../../files/probability/binomial-distribution/quiz/q2.jpg" style="width: 450px; height: 263px;"></p></div><div class="q-answer"><p>P(X = 2) = <input type="text" style="height: auto;" data-c="0.375"> <span class="review"></span></p><p>P(X = 0) = <input type="text" style="height: auto;" data-c="0.0625"> <span class="review"></span></p><p>P( <span class="math-tex">\(1\le X<4\)</span> ) = <input type="text" style="height: auto;" data-c="0.875"> <span class="review"></span></p><p>P(X > 0) = <input type="text" style="height: auto;" data-c="0.9375"> <span class="review"></span></p><p>P(X > <em><strong>a</strong></em>) = 0.6875 , <strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>P(1 <span class="math-tex">\(\le\)</span> X < 4) = P(X=1) + P(X=2) + P(X=3) = 0.25 + 0.375 + 0.25 = 0.875</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 of a head on a biased coin is 0.55.</p><p>The coin is flipped 8 times. The probability of getting 6 heads is <sup>a</sup><strong>C<sub>6</sub> x b<sup>d</sup> x 0.45<sup>e</sup></strong></p><p>Find the values of <strong><em>a</em></strong> , <strong><em>b</em></strong> , <strong><em>d</em></strong> and <strong><em>e</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><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="0.55"> <span class="review"></span></p><p><strong><em>d</em></strong> = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p><p><strong><em>e</em></strong> = <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>A binomial distribution, X has probability distribution X~B(10,0.3)</p><p>P(X = 2) =<strong> <sup>a</sup>C<sub>b</sub> x 0.3<sup>d</sup> x e<sup>8</sup> </strong></p><p>Find the values of <strong><em>a</em></strong> , <strong><em>b</em></strong> , <strong><em>d</em></strong> and <strong><em>e</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p><strong><em>d</em></strong> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p><p><strong><em>e</em></strong> = <input type="text" style="height: auto;" data-c="0.7"> <span class="review"></span></p></div><div class="q-explanation"><p><strong><em>b</em></strong><em> </em>can also take the value 8 since <sup>10</sup>C<sub>2</sub> = <sup>10</sup>C<sub>8</sub></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 class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> The random variable X has a binomial distribution with n = 12 and p = 0.35.</p><p>Calculate the following to 3 significant figures.</p></div><div class="q-answer"><ol><li>P(X = 3) = <input type="text" style="height: auto;" data-c="0.195"> <span class="review"></span></li><li>P(X <span class="math-tex">\(\le\)</span> 3) = <input type="text" style="height: auto;" data-c="0.347"> <span class="review"></span></li><li>P(X < 3) = <input type="text" style="height: auto;" data-c="0.151"> <span class="review"></span></li><li>P(X <span class="math-tex">\(\ge\)</span> 9) = <input type="text" style="height: auto;" data-c="0.00561"> <span class="review"></span></li><li>P(2<span class="math-tex">\(\le\)</span> X <span class="math-tex">\(\le\)</span> 10) = <input type="text" style="height: auto;" data-c="0.957"> <span class="review"></span></li></ol></div><div class="q-explanation"><p><img alt="" src="../../files/probability/binomial-distribution/quiz/q5.jpg" style="width: 350px; height: 192px;"></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 class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> In the ma<img alt="" height="209" src="../../files/probability/binomial-distribution/quiz/q6.png" style="float: right;" width="276">ss production of mobile phones, it is found that 5% are faulty. Phones are selected at random and put into boxes of 20. A box is selected at random.</p><p>Giving your answers to 3 significant figures find the probability that it will contain</p><hr class="hidden-separator"></div><div class="q-answer"><ol><li>Three faulty phones <input type="text" style="height: auto;" data-c="0.0596"> <span class="review"></span><hr class="hidden-separator"></li><li>No faulty phones <input type="text" style="height: auto;" data-c="0.358"> <span class="review"></span><hr class="hidden-separator"></li><li>At least two faulty phones <input type="text" style="height: auto;" data-c="0.264"> <span class="review"></span><hr class="hidden-separator"></li><li>Less than 2 faulty phones <input type="text" style="height: auto;" data-c="0.736"> <span class="review"></span><hr class="hidden-separator"></li><li>The mean number of faulty mobile phones <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></li></ol></div><div class="q-explanation"><p>X~B(20,0.05)</p><ol><li>P(X=3)</li><li>P(X=0)</li><li>P(X <span class="math-tex">\(\ge\)</span> 2)</li><li>P(X <span class="math-tex">\(\le\)</span> 1)</li><li>E(X) = np = 20x0.05</li></ol></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 class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> I<span id="cke_bm_1489C" style="display: none;"> </span>f X~B(n,0.5) and P(X<1) <span class="math-tex">\(\approx\)</span> 0.00391.</p><p>Find n</p></div><div class="q-answer"><p> n = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p>P<span id="cke_bm_1543C" style="display: none;"> </span>(X=0) <span class="math-tex">\(\approx\)</span> 0.00391</p><p><sup>n</sup>C<sub>0</sub> x 0.5<sup>0</sup> x 0.5<sup>n</sup> <span class="math-tex">\(\approx\)</span>0.00391</p><p>1 x 1 x 0.5<sup>n</sup> <span class="math-tex">\(\approx\)</span>0.00391</p><p>You can solve this equation by using logarithms, the table or graph function on your calculator or by trial and improvement</p><p>n <span class="math-tex">\(\approx \frac{\log0.00391}{\log0.5}\)</span></p><p>n = 8</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 class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> The probability that Kylie scores a penalty in a game of rugby is 0.25. Find the least number of attempts that she would need to take to be 95% sure of scoring at least one penalty.</p></div><div class="q-answer"><p>n = <input type="text" style="height: auto;" data-c="11"> <span class="review"></span></p></div><div class="q-explanation"><p>X~B(n,0.25)</p><p>P(X<span class="math-tex">\(\ge\)</span>1) > 0.95</p><p>P(X = 0) < 0.05</p><p><sup>n</sup>C<sub>0</sub> x 0.25<sup>0</sup> x 0.75<sup>n</sup> < 0.05</p><p>0.75<sup>n</sup> < 0.05</p><p>Solve 0.75<sup>n</sup> = 0.05</p><p>n =<span class="math-tex">\(\frac{\log 0.05}{\log0.75}\approx10.4\)</span></p><p>X~B(10,0.25) P(X<span class="math-tex">\(\ge\)</span>1) <span class="math-tex">\(\approx\)</span>0.944</p><p>X~B(11,0.25) P(X<span class="math-tex">\(\ge\)</span>1) <span class="math-tex">\(\approx\)</span>0.958</p><p>Therefore n = 11</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 random variable X~B(n , p) has a mean = 5 and a variance = 4</p><p>Find n</p></div><div class="q-answer"><p>n = <input type="text" style="height: auto;" data-c="0.2"> <span class="review"></span></p></div><div class="q-explanation"><p>mean = np</p><p>variance = np(1-p)</p><p>np = 5</p><p>np(1-p) = 4</p><p>substitute the first equation into the second</p><p>5(1-p) = 4</p><p>1-p = <span class="math-tex">\(\frac{4}{5}\)</span></p><p>p =<span class="math-tex">\(\frac{1}{5}\)</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><img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"> A biased coin is flipped eight times. The probability of flipping a tail is 0.6.</p><p>What is the probability of flipping <strong>at least </strong>6 tails <strong>given that </strong>we have flipped <strong>at least </strong>4 tails?</p><p>Give your answer to 3 significant figures.</p></div><div class="q-answer"><p>P = <input type="text" style="height: auto;" data-c="0.382"> <span class="review"></span></p></div><div class="q-explanation"><p>X~B(8 , 0.6)</p><p>P(X <span class="math-tex">\(\ge\)</span> 6 \ X <span class="math-tex">\(\ge\)</span> 4)</p><p><img alt="" src="../../files/probability/binomial-distribution/quiz/q10.jpg" style="width: 200px; height: 57px;"></p><p><span class="math-tex">\(\frac { P(X\ge 6) }{ P(X\ge 4) } \approx \frac { 0.3154 }{ 0.8263 } \approx 0.382\)</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> </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> <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="525"> <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/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><img alt="" src="../../files/probability/binomial-distribution/esq1.png" style="float: right; width: 100px; height: 200px;">On average, it is found that 5% of AirPods* made on a production line are faulty.</p> <div> <p>a) Find the probability that in a random sample of 10, there are</p> <p>i) No faulty AirPods</p> <p>ii) more than one faulty set of AirPods</p> <p>b) A sample of <strong><em>n </em></strong>sets of AirPods is taken from the production line. If the probability that there is at least one faulty AirPod is <strong>more than 75%,</strong> find the smallest possible value of <strong><em>n</em></strong></p> <p><strong><em>* </em></strong><em>Fictitious data. I’m sure that AirPods production is very reliable!</em></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><content> </content>a) ii) <span class="math-tex">\(P(X>1)=P(X\ge 2)\)</span></p> <p>b) Find <strong><em>n</em></strong> such that <span class="math-tex">\(P(X\ge 1)>0.75\)</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/binomial-distribution/esq1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/binomial-distribution/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="526"> <p><img alt="" src="../../files/probability/binomial-distribution/monster.jpg" style="float: right; width: 200px; height: 267px;"><img class="sibico" height="19" src="../../../img/sibico/hl-green.svg" style="vertical-align: text-bottom;" title="HL easy" width="19"> <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> <p>Monster Energy drink cans are advertised to contain 500ml. During the production, the volume <strong><em>X </em></strong>in ml in cans of the drink can be modelled by a normal distribution with mean 504 and variance 10.</p> <div> <p>a) For a randomly selected can, work out P(X > 500).</p> <p>b) Cans are put in packs of 6. Find the probability that at least 5 cans are have a volume of at least 500ml.</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>part a) of this question is about the Normal Distribution</p> <p><content>part b) is about the Binomial Distribution uses the probability found in part a)</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/binomial-distribution/esq2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/binomial-distribution/esq2.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 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="524"> <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/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><img alt="" src="../../files/probability/binomial-distribution/esq2.jpg" style="float: right; width: 250px; height: 245px;">A glass contains 5 green sweets and <strong><em>m</em></strong> sweets of other colours. A sweet is taken at random.</p> <p>a) Write down the probability that the sweet is green.</p> <p>b) The sweet is replaced in the glass and the process is repeated a further three times. Each time, it is noted whether a green sweet is taken. The variance of the number of green sweets taken over the whole process in calculated to be 0.75.</p> <p>Work out how many sweets in total there are in the glass.</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>For the Binomial distribution <span class="math-tex">\(X\sim B(n,p)\)</span></p> <p>Var(X) = <strong><em>npq q = 1 - p</em></strong><content></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/binomial-distribution/esq3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/binomial-distribution/esq3.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="692"> <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>Binomial Distribution</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" placeholder="Write your notes here..."></textarea> </div> <div class="panel-footer text-xs-center"> <span id="last-edited" class="mb-xs-3"> </span> <div class="actions mt-xs-3"> <button 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