{"id":1436,"date":"2024-05-10T16:16:36","date_gmt":"2024-05-10T20:16:36","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&#038;p=1436"},"modified":"2024-06-12T14:43:29","modified_gmt":"2024-06-12T18:43:29","slug":"binomial-distributions-more-than","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/binomial-distributions-more-than\/","title":{"raw":"Binomial Distributions - More Than","rendered":"Binomial Distributions &#8211; More Than"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nCalculate the probability of more than [latex] x[\/latex] number of successes.\r\n\r\n<\/div>\r\n<\/div>\r\nIn Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST<\/a> function, when cumulative is set to TRUE (or 1):\r\n<ul>\r\n \t<li>Excel adds up all probabilities up to and including the number of successes ([latex]x[\/latex]-value) inputted<\/li>\r\n \t<li>[latex] P(X \\le x) = \\text{BINOM.DIST}(x,n, p, 1) [\/latex]<\/li>\r\n<\/ul>\r\nWhat if we want the values above a certain x-value instead?\r\n<ul>\r\n \t<li>We need to use a complement when using Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST<\/a><\/li>\r\n \t<li>This is because [latex]P(X \\gt x) = 1 \u2212 P(X\\le x)[\/latex]<\/li>\r\n \t<li>In Excel, this gives: [latex]P(X \\gt x) = 1 \u2212 \\text{BINOM.DIST}(x, n, p, 1)[\/latex]<\/li>\r\n<\/ul>\r\n<h1>Using Formulas to Calculate More than (Example)<\/h1>\r\nIn Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST<\/a> function, when cumulative is set to TRUE (or 1), Excel adds up all probabilities up to and including the number of successes ([latex]x[\/latex]-value) inputted. What if we want the values above a certain [latex] x[\/latex]-value instead?\r\n<h2>Example 27.1 \u2013 Using the Formula<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: A hotel's records indicate that 65% of its guests are visitors from Canada.\r\n\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: From a random sample of 12 guests, what is the probability that more than 10 of them are from Canada?\r\n\r\n<span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: We know the following:\r\n<ul>\r\n \t<li>[latex]P(\\text{more than 10}) = P(x&gt;10) = P(x=11) + P(x=12) [\/latex]<\/li>\r\n \t<li>[latex]n=12[\/latex] and [latex]p=0.65[\/latex].<\/li>\r\n<\/ul>\r\nThis gives:\r\n<ul>\r\n \t<li>[latex]P(x=11) = {}_{12}C_{11} \\cdot (0.65)^{11} \\cdot (1 - 0.65)^{12-11} = \\frac{12!}{11!1!} \\cdot (0.65)^{11} \\cdot (0.35)^1 =12(0.00875)(0.35) = 0.03675 [\/latex]<\/li>\r\n \t<li>[latex]P(x=12) = {}_{12}C_{12} \\cdot (0.65)^{12} \\cdot (1 - 0.65)^{12-12} = \\frac{12!}{12!0!} \\cdot (0.65)^{12} \\cdot (0.35)^0 =1(0.00569)(1) = 0.00569 [\/latex]<\/li>\r\n \t<li>[latex]P(x\\gt10) =\u00a0 P(x=11)+P(x=12) = 0.03675+ 0.00569 = 0.04244[\/latex]<\/li>\r\n<\/ul>\r\n<span style=\"color: #003366\"><strong>Conclusion: <\/strong><\/span>There is an 4.24% chance that more than 10 of them are from Canada.\r\n<h2>Example 27.2 \u2013 Using the Formula and a Complement<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Let us now revisit example 27.1 but using a complement.\r\n\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: Use a complement to calculate the probability of more than 10 guests being from Canada.\r\n\r\n<span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: We know all of the possible outcomes need to add to 1:\r\n\r\n\\[ P(x=0)+P(x=1)+P(x=2)+P(x=3)+P(x=4)+P(x=5)+P(x=6)+P(x=7)+P(x=8)+P(x=9)+P(x=10)+P(x=11)+P(x=12) = 1 \\]\r\n\r\nWe can use the above equation to solve for [latex] P(x\\gt 10) = P(x=11) + P(x=12) [\/latex]:\r\n\r\n[latex]\r\n\r\n\\begin{align*}\r\n\r\nP(x \\gt 10) &amp;= P(x=11)+P(x=12) \\\\ \\\\\r\n\r\n&amp;= 1-[P(x=0)+P(x=1)+P(x=2)+P(x=3)+P(x=4)+P(x=5)+P(x=6)+P(x=7)+P(x=8)+P(x=9)+P(x=10)] \\\\ \\\\\r\n\r\n&amp;= 1-[0.0000+0.0001+0.0008+0.0048+0.0199+0.0591+0.1281+0.2039+0.2367+0.1954+0.1089] \\\\ \\\\\r\n\r\n&amp;= 1-0.9576 = 0.0424\r\n\r\n\\end{align*}[\/latex]\r\n\r\n<span style=\"color: #003366\"><strong>Conclusions<\/strong><\/span>:\r\n<ul>\r\n \t<li>This is quite the calculation to perform!<\/li>\r\n \t<li>If you would like more practice - try to find the probabilities given above.<\/li>\r\n \t<li>It is recommended, however, that we use Excel for calculations like these!<\/li>\r\n \t<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/Example26-3.xlsx\">Click here<\/a> to download the Excel sheet and also see the next section.<\/li>\r\n<\/ul>\r\n<h1>Using Excel to Calculate More than (Exercise)<\/h1>\r\n<ul>\r\n \t<li>Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST()<\/a> function can calculate the probability up to an x value.<\/li>\r\n \t<li>If we combine this with a complement, we can calculate the probability of more than a value.<\/li>\r\n \t<li>We will try this out in the next example.<\/li>\r\n<\/ul>\r\n<h2>Example 27.3 \u2013 Using Excel and a Complement<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Let us now revisit example 27.2 but we will use Excel.\r\n\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: Use a complement and Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST<\/a> to calculate the probability of more than 10 guests being from Canada when 12 guests are selected at random and the probability of any guest being from Canada is 65%.\r\n\r\n<span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: We know that:\r\n<ul>\r\n \t<li>P([latex]X[\/latex] \u2264 [latex]x[\/latex]) = BINOM.DIST([latex]x[\/latex], [latex]n[\/latex], [latex]p[\/latex], 1)<\/li>\r\n \t<li>P([latex]x[\/latex] &gt; 10) = 1 - P([latex]x[\/latex] \u2264 10)<\/li>\r\n<\/ul>\r\n<span style=\"color: #003366\"><strong>You try<\/strong><\/span>: Use the above to solve for P([latex]x[\/latex] &gt; 10) in the exercise below:\r\n\r\n[h5p id=\"67\"]\r\n\r\n[h5p id=\"70\"]\r\n<h1>ViDeo of Using Excel to Calculate More than (VIDEO)<\/h1>\r\n<ul>\r\n \t<li>Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST([latex]n[\/latex])<\/a> function can calculate the probability up to an x value.<\/li>\r\n \t<li>We do this by setting cumulative = TRUE or 1.<\/li>\r\n \t<li>We will try this out in the next example.<\/li>\r\n<\/ul>\r\n<h2>Example 27.4 \u2013 Using Excel and a Complement<\/h2>\r\n<span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Let us now revisit example 27.3 but let's slightly change what we want to calculate.\r\n\r\n<span style=\"color: #003366\"><strong>Question<\/strong><\/span>: What is\u00a0the probability of more than 2 guests <em><strong>NOT<\/strong><\/em> being from Canada when 12 guests are selected at random and the probability of any guest <em><strong>being<\/strong><\/em> from Canada is 65%? Use a complement and Excel's <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST<\/a>.\r\n\r\n<span style=\"color: #003366\"><strong>Solutions<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/Example27-4.xlsx\">Click here<\/a>\u00a0to download the Excel solution. Also, see the video below:\r\n\r\n[embed]https:\/\/youtu.be\/djXlwpuaQ7Q[\/embed]\r\n\r\n<span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: There is an 84.87% chance that more than 2 of the 12 guests are not from Canada.\r\n<h1>Key Takeaways (EXERCISE)<\/h1>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Takeaways: Binomial Distributions - More Than<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n[h5p id=\"73\"]\r\n\r\n[h5p id=\"74\"]\r\n\r\n<\/div>\r\n<\/div>\r\n<h1>Your Own Notes (EXERCISE)<\/h1>\r\n<ul>\r\n \t<li>Are there any notes you want to take from this section? Is there anything you'd like to copy and paste below?<\/li>\r\n \t<li>These notes are for you only (they will not be stored anywhere)<\/li>\r\n \t<li>Make sure to download them at the end to use as a reference<\/li>\r\n<\/ul>\r\n[h5p id=\"16\"]","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Calculate the probability of more than [latex]x[\/latex] number of successes.<\/p>\n<\/div>\n<\/div>\n<p>In Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST<\/a> function, when cumulative is set to TRUE (or 1):<\/p>\n<ul>\n<li>Excel adds up all probabilities up to and including the number of successes ([latex]x[\/latex]-value) inputted<\/li>\n<li>[latex]P(X \\le x) = \\text{BINOM.DIST}(x,n, p, 1)[\/latex]<\/li>\n<\/ul>\n<p>What if we want the values above a certain x-value instead?<\/p>\n<ul>\n<li>We need to use a complement when using Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST<\/a><\/li>\n<li>This is because [latex]P(X \\gt x) = 1 \u2212 P(X\\le x)[\/latex]<\/li>\n<li>In Excel, this gives: [latex]P(X \\gt x) = 1 \u2212 \\text{BINOM.DIST}(x, n, p, 1)[\/latex]<\/li>\n<\/ul>\n<h1>Using Formulas to Calculate More than (Example)<\/h1>\n<p>In Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST<\/a> function, when cumulative is set to TRUE (or 1), Excel adds up all probabilities up to and including the number of successes ([latex]x[\/latex]-value) inputted. What if we want the values above a certain [latex]x[\/latex]-value instead?<\/p>\n<h2>Example 27.1 \u2013 Using the Formula<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: A hotel&#8217;s records indicate that 65% of its guests are visitors from Canada.<\/p>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: From a random sample of 12 guests, what is the probability that more than 10 of them are from Canada?<\/p>\n<p><span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: We know the following:<\/p>\n<ul>\n<li>[latex]P(\\text{more than 10}) = P(x>10) = P(x=11) + P(x=12)[\/latex]<\/li>\n<li>[latex]n=12[\/latex] and [latex]p=0.65[\/latex].<\/li>\n<\/ul>\n<p>This gives:<\/p>\n<ul>\n<li>[latex]P(x=11) = {}_{12}C_{11} \\cdot (0.65)^{11} \\cdot (1 - 0.65)^{12-11} = \\frac{12!}{11!1!} \\cdot (0.65)^{11} \\cdot (0.35)^1 =12(0.00875)(0.35) = 0.03675[\/latex]<\/li>\n<li>[latex]P(x=12) = {}_{12}C_{12} \\cdot (0.65)^{12} \\cdot (1 - 0.65)^{12-12} = \\frac{12!}{12!0!} \\cdot (0.65)^{12} \\cdot (0.35)^0 =1(0.00569)(1) = 0.00569[\/latex]<\/li>\n<li>[latex]P(x\\gt10) =\u00a0 P(x=11)+P(x=12) = 0.03675+ 0.00569 = 0.04244[\/latex]<\/li>\n<\/ul>\n<p><span style=\"color: #003366\"><strong>Conclusion: <\/strong><\/span>There is an 4.24% chance that more than 10 of them are from Canada.<\/p>\n<h2>Example 27.2 \u2013 Using the Formula and a Complement<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Let us now revisit example 27.1 but using a complement.<\/p>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: Use a complement to calculate the probability of more than 10 guests being from Canada.<\/p>\n<p><span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: We know all of the possible outcomes need to add to 1:<\/p>\n<p>\\[ P(x=0)+P(x=1)+P(x=2)+P(x=3)+P(x=4)+P(x=5)+P(x=6)+P(x=7)+P(x=8)+P(x=9)+P(x=10)+P(x=11)+P(x=12) = 1 \\]<\/p>\n<p>We can use the above equation to solve for [latex]P(x\\gt 10) = P(x=11) + P(x=12)[\/latex]:<\/p>\n<p>[latex]\\begin{align*}    P(x \\gt 10) &= P(x=11)+P(x=12) \\\\ \\\\    &= 1-[P(x=0)+P(x=1)+P(x=2)+P(x=3)+P(x=4)+P(x=5)+P(x=6)+P(x=7)+P(x=8)+P(x=9)+P(x=10)] \\\\ \\\\    &= 1-[0.0000+0.0001+0.0008+0.0048+0.0199+0.0591+0.1281+0.2039+0.2367+0.1954+0.1089] \\\\ \\\\    &= 1-0.9576 = 0.0424    \\end{align*}[\/latex]<\/p>\n<p><span style=\"color: #003366\"><strong>Conclusions<\/strong><\/span>:<\/p>\n<ul>\n<li>This is quite the calculation to perform!<\/li>\n<li>If you would like more practice &#8211; try to find the probabilities given above.<\/li>\n<li>It is recommended, however, that we use Excel for calculations like these!<\/li>\n<li><a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/Example26-3.xlsx\">Click here<\/a> to download the Excel sheet and also see the next section.<\/li>\n<\/ul>\n<h1>Using Excel to Calculate More than (Exercise)<\/h1>\n<ul>\n<li>Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST()<\/a> function can calculate the probability up to an x value.<\/li>\n<li>If we combine this with a complement, we can calculate the probability of more than a value.<\/li>\n<li>We will try this out in the next example.<\/li>\n<\/ul>\n<h2>Example 27.3 \u2013 Using Excel and a Complement<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Let us now revisit example 27.2 but we will use Excel.<\/p>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: Use a complement and Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST<\/a> to calculate the probability of more than 10 guests being from Canada when 12 guests are selected at random and the probability of any guest being from Canada is 65%.<\/p>\n<p><span style=\"color: #003366\"><strong>Solution<\/strong><\/span>: We know that:<\/p>\n<ul>\n<li>P([latex]X[\/latex] \u2264 [latex]x[\/latex]) = BINOM.DIST([latex]x[\/latex], [latex]n[\/latex], [latex]p[\/latex], 1)<\/li>\n<li>P([latex]x[\/latex] &gt; 10) = 1 &#8211; P([latex]x[\/latex] \u2264 10)<\/li>\n<\/ul>\n<p><span style=\"color: #003366\"><strong>You try<\/strong><\/span>: Use the above to solve for P([latex]x[\/latex] &gt; 10) in the exercise below:<\/p>\n<div id=\"h5p-67\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-67\" class=\"h5p-iframe\" data-content-id=\"67\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 26.3.3 \u2013 Binomial - Using Excel and a Complement\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-70\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-70\" class=\"h5p-iframe\" data-content-id=\"70\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Example 26.3.3 \u2013 Using Excel and a Complement Solution\"><\/iframe><\/div>\n<\/div>\n<h1>ViDeo of Using Excel to Calculate More than (VIDEO)<\/h1>\n<ul>\n<li>Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST([latex]n[\/latex])<\/a> function can calculate the probability up to an x value.<\/li>\n<li>We do this by setting cumulative = TRUE or 1.<\/li>\n<li>We will try this out in the next example.<\/li>\n<\/ul>\n<h2>Example 27.4 \u2013 Using Excel and a Complement<\/h2>\n<p><span style=\"color: #003366\"><strong>Problem Setup<\/strong><\/span>: Let us now revisit example 27.3 but let&#8217;s slightly change what we want to calculate.<\/p>\n<p><span style=\"color: #003366\"><strong>Question<\/strong><\/span>: What is\u00a0the probability of more than 2 guests <em><strong>NOT<\/strong><\/em> being from Canada when 12 guests are selected at random and the probability of any guest <em><strong>being<\/strong><\/em> from Canada is 65%? Use a complement and Excel&#8217;s <a href=\"https:\/\/support.microsoft.com\/en-us\/office\/binomdist-function-506a663e-c4ca-428d-b9a8-05583d68789c#:~:text=Returns%20the%20individual%20term%20binomial,is%20constant%20throughout%20the%20experiment.\">BINOM.DIST<\/a>.<\/p>\n<p><span style=\"color: #003366\"><strong>Solutions<\/strong><\/span>: <a href=\"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-content\/uploads\/sites\/2128\/2024\/05\/Example27-4.xlsx\">Click here<\/a>\u00a0to download the Excel solution. Also, see the video below:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"How to calculate &#39;more than&#39; using Excel&#39;s Binom.Dist() function\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/djXlwpuaQ7Q?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><span style=\"color: #003366\"><strong>Conclusion<\/strong><\/span>: There is an 84.87% chance that more than 2 of the 12 guests are not from Canada.<\/p>\n<h1>Key Takeaways (EXERCISE)<\/h1>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways: Binomial Distributions &#8211; More Than<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<div id=\"h5p-73\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-73\" class=\"h5p-iframe\" data-content-id=\"73\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key Takeaways for Binomial Distributions - More Than\"><\/iframe><\/div>\n<\/div>\n<div id=\"h5p-74\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-74\" class=\"h5p-iframe\" data-content-id=\"74\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key Takeaways for Binomial Distributions - More Than - Solutions\"><\/iframe><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h1>Your Own Notes (EXERCISE)<\/h1>\n<ul>\n<li>Are there any notes you want to take from this section? Is there anything you&#8217;d like to copy and paste below?<\/li>\n<li>These notes are for you only (they will not be stored anywhere)<\/li>\n<li>Make sure to download them at the end to use as a reference<\/li>\n<\/ul>\n<div id=\"h5p-16\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-16\" class=\"h5p-iframe\" data-content-id=\"16\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Key takeaways, notes and comments from this section document tool.\"><\/iframe><\/div>\n<\/div>\n","protected":false},"author":865,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1436","chapter","type-chapter","status-publish","hentry"],"part":231,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1436","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/users\/865"}],"version-history":[{"count":21,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1436\/revisions"}],"predecessor-version":[{"id":1955,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1436\/revisions\/1955"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/parts\/231"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapters\/1436\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/media?parent=1436"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1436"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/contributor?post=1436"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/wp-json\/wp\/v2\/license?post=1436"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}