{"id":1035,"date":"2023-06-22T01:45:13","date_gmt":"2023-06-22T01:45:13","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/binomial-distribution-learn-it-4\/"},"modified":"2023-11-09T19:08:39","modified_gmt":"2023-11-09T19:08:39","slug":"binomial-distribution-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/binomial-distribution-learn-it-4\/","title":{"raw":"Binomial Distribution: Learn It 4","rendered":"Binomial Distribution: Learn It 4"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Use a binomial distribution to calculate probability<\/li>\r\n\t<li>Determine if a probability model meets the conditions for a binomial distribution<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h3>Binomial Distribution Formula<\/h3>\r\n<section class=\"textbox example\">Let\u2019s revisit the experiment flipping the coin [latex]3[\/latex] times and counting the number of tails obtained.<br \/>\r\n[reveal-answer q=\"225721\"]Outcomes of flipping a coin [latex]3[\/latex] times.[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"225721\"]Recall that the outcomes of the experiment are as given in the following table:\r\n\r\n<div align=\"center\">\r\n<table style=\"height: 108px; width: 386px;\">\r\n<tbody>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"height: 12px; width: 118.547px;\">\r\n<p style=\"text-align: center;\"><strong>Experimental <\/strong><\/p>\r\n<p style=\"text-align: center;\"><strong>Outcome<\/strong><\/p>\r\n<\/td>\r\n<td style=\"height: 12px; width: 243.766px;\">\r\n<p style=\"text-align: center;\"><strong>[latex]X[\/latex]<\/strong><\/p>\r\n<p style=\"text-align: center;\"><strong>Number of Tails in 3 Flips of a Coin<\/strong><\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HHH<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HHT<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HTH<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">THH<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">TTH<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">THT<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HTT<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 12px;\">\r\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">TTT<\/td>\r\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/div>\r\n<p>Notice that the number of ways to obtain each number of successes in a binomial experiment increases pretty quickly. If we were to flip [latex]4[\/latex] coins, there would be:<\/p>\r\n<ul>\r\n\t<li>[latex]1[\/latex] way to obtain [latex]0[\/latex] tails<\/li>\r\n\t<li>[latex]4[\/latex] ways to obtain [latex]1[\/latex] tail<\/li>\r\n\t<li>[latex]6[\/latex] ways to obtain [latex]2[\/latex] tails<\/li>\r\n\t<li>[latex]4[\/latex] ways to obtain [latex]3[\/latex] tails<\/li>\r\n\t<li>[latex]1[\/latex] way to obtain [latex]4[\/latex] tails<\/li>\r\n<\/ul>\r\n<p>There is a formula that lets us compute these probabilities more easily.<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>binomial distribution formula<\/h3>\r\n<p>For a binomial experiment in which the probability of success is [latex]p[\/latex] and there are [latex]n[\/latex] trials, the <strong>binomial distribution<\/strong> gives the probability of obtaining [latex]x[\/latex] successes is<\/p>\r\n<p style=\"text-align: center;\">[latex] P(X=x) = \\dfrac{n!}{x!(n-x)!} \\cdot p^{x} \\cdot (1-p)^{n-x} [\/latex]<\/p>\r\n<p>where [latex] \\frac{n!}{x!(n-x)!} [\/latex] is called \u201c[latex]n \\mbox{ choose } x[\/latex],\u201d which computes the number of ways to obtain [latex] x [\/latex] successes out of [latex] n [\/latex] trials.<\/p>\r\n<\/section>\r\n<p>The exclamation mark is the symbol for a factorial. You won\u2019t need to calculate this because we will be using technology for our computations, but [latex] n! [\/latex] is the product of all the positive numbers preceding the number [latex]n[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex] n! = n(n-1)(n-2) \\cdots (2)(1) [\/latex]<\/p>\r\n<p>For example, [latex] 3! =(3)(2)(1) = 6 [\/latex].<\/p>\r\n<section class=\"textbox example\">On the experiment flipping the coin [latex]3[\/latex] times and counting the number of tails obtained, we found that there were [latex]3[\/latex] ways to obtain [latex]1[\/latex] tail in [latex]3[\/latex] coin flips.To see how this corresponds to the formula, observe that for [latex]n=3[\/latex] and [latex]x=1[\/latex]:[latex]3\\text{ choose }1 = \\frac{3!}{1!(3-1)!} = \\frac{3!}{1!2!} = \\frac{6}{1 \\cdot 2} = \\frac{6}{2} = 3 [\/latex]<\/section>\r\n<p>As mentioned, we will be using technology to compute these probabilities, so you won\u2019t need to worry much about the formula.<\/p>\r\n<p>Feel free to explore the tool.<\/p>\r\n<p>You can click on the <strong>Find Probabilities<\/strong> tab, input values for [latex]n[\/latex], [latex]p[\/latex], and [latex]x[\/latex], and then select which type of probability you would like to compute from the drop-down menu.<\/p>\r\n<p><iframe src=\"https:\/\/lumen-learning.shinyapps.io\/binomialdist\/\" width=\"100%\" height=\"900\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/binomialdist\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use a binomial distribution to calculate probability<\/li>\n<li>Determine if a probability model meets the conditions for a binomial distribution<\/li>\n<\/ul>\n<\/section>\n<h3>Binomial Distribution Formula<\/h3>\n<section class=\"textbox example\">Let\u2019s revisit the experiment flipping the coin [latex]3[\/latex] times and counting the number of tails obtained.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q225721\">Outcomes of flipping a coin [latex]3[\/latex] times.<\/button><\/p>\n<div id=\"q225721\" class=\"hidden-answer\" style=\"display: none\">Recall that the outcomes of the experiment are as given in the following table:<\/p>\n<div style=\"margin: auto;\">\n<table style=\"height: 108px; width: 386px;\">\n<tbody>\n<tr style=\"height: 12px;\">\n<td style=\"height: 12px; width: 118.547px;\">\n<p style=\"text-align: center;\"><strong>Experimental <\/strong><\/p>\n<p style=\"text-align: center;\"><strong>Outcome<\/strong><\/p>\n<\/td>\n<td style=\"height: 12px; width: 243.766px;\">\n<p style=\"text-align: center;\"><strong>[latex]X[\/latex]<\/strong><\/p>\n<p style=\"text-align: center;\"><strong>Number of Tails in 3 Flips of a Coin<\/strong><\/p>\n<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HHH<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]0[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HHT<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HTH<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">THH<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]1[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">TTH<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">THT<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">HTT<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]2[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 12px;\">\n<td style=\"text-align: center; height: 12px; width: 118.547px;\">TTT<\/td>\n<td style=\"text-align: center; height: 12px; width: 243.766px;\">[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that the number of ways to obtain each number of successes in a binomial experiment increases pretty quickly. If we were to flip [latex]4[\/latex] coins, there would be:<\/p>\n<ul>\n<li>[latex]1[\/latex] way to obtain [latex]0[\/latex] tails<\/li>\n<li>[latex]4[\/latex] ways to obtain [latex]1[\/latex] tail<\/li>\n<li>[latex]6[\/latex] ways to obtain [latex]2[\/latex] tails<\/li>\n<li>[latex]4[\/latex] ways to obtain [latex]3[\/latex] tails<\/li>\n<li>[latex]1[\/latex] way to obtain [latex]4[\/latex] tails<\/li>\n<\/ul>\n<p>There is a formula that lets us compute these probabilities more easily.<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>binomial distribution formula<\/h3>\n<p>For a binomial experiment in which the probability of success is [latex]p[\/latex] and there are [latex]n[\/latex] trials, the <strong>binomial distribution<\/strong> gives the probability of obtaining [latex]x[\/latex] successes is<\/p>\n<p style=\"text-align: center;\">[latex]P(X=x) = \\dfrac{n!}{x!(n-x)!} \\cdot p^{x} \\cdot (1-p)^{n-x}[\/latex]<\/p>\n<p>where [latex]\\frac{n!}{x!(n-x)!}[\/latex] is called \u201c[latex]n \\mbox{ choose } x[\/latex],\u201d which computes the number of ways to obtain [latex]x[\/latex] successes out of [latex]n[\/latex] trials.<\/p>\n<\/section>\n<p>The exclamation mark is the symbol for a factorial. You won\u2019t need to calculate this because we will be using technology for our computations, but [latex]n![\/latex] is the product of all the positive numbers preceding the number [latex]n[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]n! = n(n-1)(n-2) \\cdots (2)(1)[\/latex]<\/p>\n<p>For example, [latex]3! =(3)(2)(1) = 6[\/latex].<\/p>\n<section class=\"textbox example\">On the experiment flipping the coin [latex]3[\/latex] times and counting the number of tails obtained, we found that there were [latex]3[\/latex] ways to obtain [latex]1[\/latex] tail in [latex]3[\/latex] coin flips.To see how this corresponds to the formula, observe that for [latex]n=3[\/latex] and [latex]x=1[\/latex]:[latex]3\\text{ choose }1 = \\frac{3!}{1!(3-1)!} = \\frac{3!}{1!2!} = \\frac{6}{1 \\cdot 2} = \\frac{6}{2} = 3[\/latex]<\/section>\n<p>As mentioned, we will be using technology to compute these probabilities, so you won\u2019t need to worry much about the formula.<\/p>\n<p>Feel free to explore the tool.<\/p>\n<p>You can click on the <strong>Find Probabilities<\/strong> tab, input values for [latex]n[\/latex], [latex]p[\/latex], and [latex]x[\/latex], and then select which type of probability you would like to compute from the drop-down menu.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/lumen-learning.shinyapps.io\/binomialdist\/\" width=\"100%\" height=\"900\" frameborder=\"no\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p>[<a href=\"https:\/\/lumen-learning.shinyapps.io\/binomialdist\/\" target=\"_blank\" rel=\"noopener\">Trouble viewing? Click to open in a new tab.<\/a>]<\/p>\n","protected":false},"author":8,"menu_order":14,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":2912,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1035"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1035\/revisions"}],"predecessor-version":[{"id":4313,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1035\/revisions\/4313"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/2912"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1035\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1035"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1035"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1035"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1035"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}