{"id":1240,"date":"2021-08-20T17:55:39","date_gmt":"2021-08-20T17:55:39","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1240"},"modified":"2022-01-31T18:46:56","modified_gmt":"2022-01-31T18:46:56","slug":"hypergeometric-distribution-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/hypergeometric-distribution-2\/","title":{"raw":"Hypergeometric Probability Distribution Function","rendered":"Hypergeometric Probability Distribution Function"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>State hypergeometric probabilities using mathematical notation<\/li>\r\n \t<li>Calculate the mean of a hypergeometric random variable<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h3 data-type=\"title\">Notation for the Hypergeometric: H = Hypergeometric Probability Distribution Function<\/h3>\r\n[latex]X[\/latex] ~ [latex]H(r{,} b{,} n)[\/latex]\r\n\r\nRead this as \"[latex]X[\/latex]\u00a0is a random variable with a hypergeometric distribution.\" The parameters are\u00a0[latex]r[\/latex],\u00a0[latex]b[\/latex], and\u00a0[latex]n[\/latex];\u00a0[latex]r[\/latex]\u00a0= the size of the group of interest (first group),\u00a0[latex]b[\/latex]\u00a0= the size of the second group,\u00a0[latex]n[\/latex]\u00a0= the size of the chosen sample.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nA school site committee is to be chosen randomly from six men and five women. If the committee consists of four members chosen randomly, what is the probability that two of them are men? How many men do you expect to be on the committee?\r\n\r\n[reveal-answer q=\"640852\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"640852\"]\r\n<p id=\"element-875\" class=\" \">Let\u00a0[latex]X[\/latex]\u00a0= the number of men on the committee of four. The men are the group of interest (first group).<\/p>\r\n<p id=\"element-876\" class=\" \">[latex]X[\/latex] takes on the values 0, 1, 2, 3, 4, where\u00a0[latex]r[\/latex]<em data-effect=\"italics\">\u00a0= 6<\/em>,\u00a0[latex]b[\/latex]<em data-effect=\"italics\">\u00a0= 5<\/em>, and\u00a0[latex]n[\/latex]<em data-effect=\"italics\">\u00a0= 4<\/em>.\u00a0[latex]X[\/latex]\u00a0<em data-effect=\"italics\">~ <\/em>[latex]H[\/latex](6, 5, 4)<\/p>\r\n<p id=\"element-877\" class=\"finger \">Find P(x\u00a0= 2). P(x = 2) = 0.4545 (calculator or computer)<\/p>\r\n\r\n<div id=\"id43780917\" class=\"finger ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<h4 id=\"4\" data-type=\"title\">NOTE<\/h4>\r\n<\/header><section>\r\n<p id=\"fs-idm9108464\" class=\" \">Currently, the TI-83+ and TI-84 do not have hypergeometric probability functions. There are a number of computer packages, including Microsoft Excel, that do.<\/p>\r\n&nbsp;\r\n\r\n<\/section><\/div>\r\n<p id=\"element-879\" class=\" \">The probability that there are two men on the committee is about 0.45.<\/p>\r\n<p id=\"element-136\" class=\" \">The graph of\u00a0[latex]X[\/latex]\u00a0~\u00a0[latex]H[\/latex](6, 5, 4) is:<\/p>\r\n\r\n<div id=\"fs-idp36734864\" class=\"os-figure\">\r\n<figure data-id=\"fs-idp36734864\"><span id=\"id43780967\" data-type=\"media\" data-display=\"block\" data-alt=\"This graph shows a hypergeometric probability distribution. It has five bars that are slightly normally distributed. The x-axis shows values from 0 to 4 in increments of 1, representing the number of men on the four-person committee. The y-axis ranges from 0 to 0.5 in increments of 0.1.\"><\/span><\/figure>\r\n<div><img class=\"aligncenter wp-image-462 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/30171407\/a935e6e12aae78185dc62986769ffab118f46c831.jpeg\" alt=\"This graph shows a hypergeometric probability distribution. It has five bars that are slightly normally distributed. The x-axis shows values from 0 to 4 in increments of 1, representing the number of men on the four-person committee. The y-axis ranges from 0 to 0.5 in increments of 0.1.\" width=\"492\" height=\"312\" \/><\/div>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure\u00a0<\/span><span class=\"os-number\">4.4<\/span><\/div>\r\n<\/div>\r\n<p id=\"element-704\" class=\" \">The\u00a0[latex]y[\/latex]-axis contains the probability of\u00a0[latex]X[\/latex], where\u00a0[latex]X[\/latex]\u00a0= the number of men on the committee.<\/p>\r\n<p id=\"element-369\" class=\" \">You would expect\u00a0[latex]m[\/latex]\u00a0= 2.18 (about two) men on the committee.<\/p>\r\n<p id=\"element-702\" class=\" \">The formula for the mean is [latex] \\mu = \\frac{nr}{r+b} = \\frac{(4)(6)}{6+5} = 2.18[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nAn intramural basketball team is to be chosen randomly from 15 boys and 12 girls. The team has ten slots. You want to know the probability that eight of the players will be boys. What is the group of interest and the sample?\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>State hypergeometric probabilities using mathematical notation<\/li>\n<li>Calculate the mean of a hypergeometric random variable<\/li>\n<\/ul>\n<\/div>\n<h3 data-type=\"title\">Notation for the Hypergeometric: H = Hypergeometric Probability Distribution Function<\/h3>\n<p>[latex]X[\/latex] ~ [latex]H(r{,} b{,} n)[\/latex]<\/p>\n<p>Read this as &#8220;[latex]X[\/latex]\u00a0is a random variable with a hypergeometric distribution.&#8221; The parameters are\u00a0[latex]r[\/latex],\u00a0[latex]b[\/latex], and\u00a0[latex]n[\/latex];\u00a0[latex]r[\/latex]\u00a0= the size of the group of interest (first group),\u00a0[latex]b[\/latex]\u00a0= the size of the second group,\u00a0[latex]n[\/latex]\u00a0= the size of the chosen sample.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>A school site committee is to be chosen randomly from six men and five women. If the committee consists of four members chosen randomly, what is the probability that two of them are men? How many men do you expect to be on the committee?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q640852\">Show Answer<\/span><\/p>\n<div id=\"q640852\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"element-875\" class=\"\">Let\u00a0[latex]X[\/latex]\u00a0= the number of men on the committee of four. The men are the group of interest (first group).<\/p>\n<p id=\"element-876\" class=\"\">[latex]X[\/latex] takes on the values 0, 1, 2, 3, 4, where\u00a0[latex]r[\/latex]<em data-effect=\"italics\">\u00a0= 6<\/em>,\u00a0[latex]b[\/latex]<em data-effect=\"italics\">\u00a0= 5<\/em>, and\u00a0[latex]n[\/latex]<em data-effect=\"italics\">\u00a0= 4<\/em>.\u00a0[latex]X[\/latex]\u00a0<em data-effect=\"italics\">~ <\/em>[latex]H[\/latex](6, 5, 4)<\/p>\n<p id=\"element-877\" class=\"finger\">Find P(x\u00a0= 2). P(x = 2) = 0.4545 (calculator or computer)<\/p>\n<div id=\"id43780917\" class=\"finger ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<h4 id=\"4\" data-type=\"title\">NOTE<\/h4>\n<\/header>\n<section>\n<p id=\"fs-idm9108464\" class=\"\">Currently, the TI-83+ and TI-84 do not have hypergeometric probability functions. There are a number of computer packages, including Microsoft Excel, that do.<\/p>\n<p>&nbsp;<\/p>\n<\/section>\n<\/div>\n<p id=\"element-879\" class=\"\">The probability that there are two men on the committee is about 0.45.<\/p>\n<p id=\"element-136\" class=\"\">The graph of\u00a0[latex]X[\/latex]\u00a0~\u00a0[latex]H[\/latex](6, 5, 4) is:<\/p>\n<div id=\"fs-idp36734864\" class=\"os-figure\">\n<figure data-id=\"fs-idp36734864\"><span id=\"id43780967\" data-type=\"media\" data-display=\"block\" data-alt=\"This graph shows a hypergeometric probability distribution. It has five bars that are slightly normally distributed. The x-axis shows values from 0 to 4 in increments of 1, representing the number of men on the four-person committee. The y-axis ranges from 0 to 0.5 in increments of 0.1.\"><\/span><\/figure>\n<div><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-462 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5668\/2021\/07\/30171407\/a935e6e12aae78185dc62986769ffab118f46c831.jpeg\" alt=\"This graph shows a hypergeometric probability distribution. It has five bars that are slightly normally distributed. The x-axis shows values from 0 to 4 in increments of 1, representing the number of men on the four-person committee. The y-axis ranges from 0 to 0.5 in increments of 0.1.\" width=\"492\" height=\"312\" \/><\/div>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure\u00a0<\/span><span class=\"os-number\">4.4<\/span><\/div>\n<\/div>\n<p id=\"element-704\" class=\"\">The\u00a0[latex]y[\/latex]-axis contains the probability of\u00a0[latex]X[\/latex], where\u00a0[latex]X[\/latex]\u00a0= the number of men on the committee.<\/p>\n<p id=\"element-369\" class=\"\">You would expect\u00a0[latex]m[\/latex]\u00a0= 2.18 (about two) men on the committee.<\/p>\n<p id=\"element-702\" class=\"\">The formula for the mean is [latex]\\mu = \\frac{nr}{r+b} = \\frac{(4)(6)}{6+5} = 2.18[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>An intramural basketball team is to be chosen randomly from 15 boys and 12 girls. The team has ten slots. You want to know the probability that eight of the players will be boys. What is the group of interest and the sample?<\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1240\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Hypergeometric Distribution. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/4-5-hypergeometric-distribution\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/4-5-hypergeometric-distribution<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":23,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Hypergeometric Distribution\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/4-5-hypergeometric-distribution\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1240","chapter","type-chapter","status-publish","hentry"],"part":240,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1240","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":5,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1240\/revisions"}],"predecessor-version":[{"id":3579,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1240\/revisions\/3579"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/240"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1240\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1240"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1240"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1240"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1240"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}