{"id":1230,"date":"2021-08-20T17:26:01","date_gmt":"2021-08-20T17:26:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1230"},"modified":"2023-12-05T09:10:30","modified_gmt":"2023-12-05T09:10:30","slug":"geometric-distribution-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/geometric-distribution-2\/","title":{"raw":"Geometric Probability Distribution Function","rendered":"Geometric Probability Distribution Function"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<section>\r\n<ul id=\"list14235\">\r\n \t<li>State geometric probabilities using mathematical notation<\/li>\r\n \t<li>Calculate the mean and standard deviation of a geometric random variable<\/li>\r\n \t<li>Calculate a geometric probability using technology<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<h2>Notation for the Geometric: [latex]G=[\/latex] Geometric Probability Distribution Function<\/h2>\r\n<p style=\"text-align: center;\">[latex]X{\\sim}G(p)[\/latex]<\/p>\r\nRead this as \"[latex]X[\/latex] is a random variable with a <strong>geometric distribution<\/strong>.\" The parameter is [latex]p[\/latex]; [latex]p=[\/latex] the probability of a success for each trial.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nAssume that the probability of a defective computer component is 0.02. Components are randomly selected. Find the probability that the first defect is caused by the seventh component tested. How many components do you expect to test until one is found to be defective?\r\n\r\n[reveal-answer q=\"607620\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"607620\"]\r\n\r\nLet [latex]X=[\/latex] the number of computer components tested until the first defect is found.\r\n\r\n[latex]X[\/latex] takes on the values 1, 2, 3, ... where [latex]p = 0.02[\/latex].\r\n<p style=\"text-align: center;\">[latex]X{\\sim}G(0.02)[\/latex]<\/p>\r\nFind [latex]P(x=7)[\/latex]. [latex]P(x=7)=0.0177[\/latex].\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\">USING THE TI-83, 83+, 84, 84+ CALCULATOR<\/span><\/h2>\r\nTo find the probability that [latex]x=7[\/latex],\r\n<ul>\r\n \t<li>Enter 2nd, DISTR<\/li>\r\n \t<li>Scroll down and select geometpdf(<\/li>\r\n \t<li>Press ENTER<\/li>\r\n \t<li>Enter 0.02, 7); press ENTER to see the result: [latex]P(x=7)=0.0177[\/latex]<\/li>\r\n<\/ul>\r\nTo find the probability that [latex]x\\leq7[\/latex], follow the same instructions EXCEPT select E:geometcdf(as the distribution function.\r\n\r\nThe probability that the seventh component is the first defect is 0.0177.\r\n\r\nThe graph of [latex]X{\\sim}G(0.02)[\/latex] is:\r\n\r\n<img src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/w63c-7tuw327i#fixme#fixme#fixme\" alt=\"This graph shows a geometric probability distribution. It consists of bars that peak at the left and slope downwards with each successive bar to the right. The values on the x-axis count the number of computer components tested until the defect is found. The y-axis is scaled from 0 to 0.02 in increments of 0.005.\" \/>\r\n\r\nThe\u00a0[latex]y[\/latex]-axis contains the probability of [latex]x[\/latex], where [latex]X=[\/latex] the number of computer components tested.\r\n\r\nThe number of components that you would expect to test until you find the first defective one is the mean,\u00a0[latex]\\displaystyle{\\mu}={50}[\/latex].\r\n\r\nThe formula for the mean is\u00a0[latex]\\displaystyle{\\mu}=\\frac{{1}}{{p}}=\\frac{{1}}{{0.02}}={50}[\/latex]\r\n\r\nThe formula for the variance is\u00a0[latex]\\displaystyle{\\sigma}^{{2}}={(\\frac{{1}}{{p}})}{(\\frac{{1}}{{p}}-{1})}={(\\frac{{1}}{{0.02}})}{(\\frac{{1}}{{0.02}}-{1})}={2},{450}[\/latex]\r\n\r\nThe standard deviation is\u00a0[latex]\\displaystyle{\\sigma}=\\sqrt{{{(\\frac{{1}}{{p}})}{(\\frac{{1}}{{p}}-{1})}}}=\\sqrt{{{(\\frac{{1}}{{0.02}})}{(\\frac{{1}}{{0.02}}-{1})}}}={49.5}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nThe probability of a defective steel rod is 0.01. Steel rods are selected at random. Find the probability that the first defect occurs on the ninth steel rod. Use the TI-83+ or TI-84 calculator to find the answer.\r\n\r\n[reveal-answer q=\"787284\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"787284\"][latex]P(x=9)=0.0092[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe lifetime risk of developing pancreatic cancer is about one in 78 (1.28%). Let [latex]X=[\/latex]the number of people you ask until one says he or she has pancreatic cancer. Then [latex]X[\/latex] is a discrete random variable with a geometric distribution: [latex]\\displaystyle{X}~{G}{(\\frac{{1}}{{78}})}{\\quad\\text{or}\\quad}{X}~{G}{({0.0128})}[\/latex]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>What is the probability of that you ask ten people before one says he or she has pancreatic cancer?<\/li>\r\n \t<li>What is the probability that you must ask 20 people?<\/li>\r\n \t<li>Find the (i) mean and (ii) standard deviation of [latex]X[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"556559\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"556559\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]P(x=10)=\\text{geometpdf}(0.0128,10)=0.0114[\/latex]<\/li>\r\n \t<li>[latex]P(x=20)=\\text{geometpdf}(0.0128,20)=0.01[\/latex]\r\n<ol style=\"list-style-type: lower-roman;\">\r\n \t<li>[latex]\\text{Mean}={\\mu}=\\frac{{1}}{{p}}=\\frac{{1}}{{0.0128}}={78}[\/latex]<\/li>\r\n \t<li>[latex]\\text{Standard Deviation}={\\sigma}=\\sqrt{{\\frac{{{1}-{p}}}{{{p}^{{2}}}}}}=\\sqrt{{\\frac{{{1}-{0.0128}}}{{0.0128}^{{2}}}}}\\approx{77.6234}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nThe literacy rate for a nation measures the proportion of people age 15 and over who can read and write. The literacy rate for women in Afghanistan is 12%. Let\u00a0[latex]X=[\/latex] the number of Afghani women you ask until one says that she is literate.\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>What is the probability distribution of [latex]X[\/latex]?<\/li>\r\n \t<li>What is the probability that you ask five women before one says she is literate?<\/li>\r\n \t<li>What is the probability that you must ask ten women?<\/li>\r\n \t<li>Find the (i) mean and (ii) standard deviation of [latex]X[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"935609\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"935609\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]X{\\sim}G(0.12)[\/latex]<\/li>\r\n \t<li>[latex]P(x=5)=\\text{geometpdf}(0.12,5)=0.0720[\/latex]<\/li>\r\n \t<li>[latex]P(x=10)=\\text{geometpdf}(0.12,10)=0.0380[\/latex]\r\n<ol style=\"list-style-type: lower-roman;\">\r\n \t<li>[latex]\\text{Mean}={\\mu}=\\frac{{1}}{{p}}=\\frac{{1}}{{0.12}}\\approx{3333}[\/latex]<\/li>\r\n \t<li>[latex]\\text{Standard Deviation}={\\sigma}=\\sqrt{{\\frac{{{1}-{p}}}{{{p}^{{2}}}}}}=\\sqrt{{\\frac{{{1}-{0.12}}}{{{0.12}^{{2}}}}}}\\approx{7.8174}[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<section>\n<ul id=\"list14235\">\n<li>State geometric probabilities using mathematical notation<\/li>\n<li>Calculate the mean and standard deviation of a geometric random variable<\/li>\n<li>Calculate a geometric probability using technology<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<h2>Notation for the Geometric: [latex]G=[\/latex] Geometric Probability Distribution Function<\/h2>\n<p style=\"text-align: center;\">[latex]X{\\sim}G(p)[\/latex]<\/p>\n<p>Read this as &#8220;[latex]X[\/latex] is a random variable with a <strong>geometric distribution<\/strong>.&#8221; The parameter is [latex]p[\/latex]; [latex]p=[\/latex] the probability of a success for each trial.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Assume that the probability of a defective computer component is 0.02. Components are randomly selected. Find the probability that the first defect is caused by the seventh component tested. How many components do you expect to test until one is found to be defective?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q607620\">Show Answer<\/span><\/p>\n<div id=\"q607620\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let [latex]X=[\/latex] the number of computer components tested until the first defect is found.<\/p>\n<p>[latex]X[\/latex] takes on the values 1, 2, 3, &#8230; where [latex]p = 0.02[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]X{\\sim}G(0.02)[\/latex]<\/p>\n<p>Find [latex]P(x=7)[\/latex]. [latex]P(x=7)=0.0177[\/latex].<\/p>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\">USING THE TI-83, 83+, 84, 84+ CALCULATOR<\/span><\/h2>\n<p>To find the probability that [latex]x=7[\/latex],<\/p>\n<ul>\n<li>Enter 2nd, DISTR<\/li>\n<li>Scroll down and select geometpdf(<\/li>\n<li>Press ENTER<\/li>\n<li>Enter 0.02, 7); press ENTER to see the result: [latex]P(x=7)=0.0177[\/latex]<\/li>\n<\/ul>\n<p>To find the probability that [latex]x\\leq7[\/latex], follow the same instructions EXCEPT select E:geometcdf(as the distribution function.<\/p>\n<p>The probability that the seventh component is the first defect is 0.0177.<\/p>\n<p>The graph of [latex]X{\\sim}G(0.02)[\/latex] is:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/w63c-7tuw327i#fixme#fixme#fixme\" alt=\"This graph shows a geometric probability distribution. It consists of bars that peak at the left and slope downwards with each successive bar to the right. The values on the x-axis count the number of computer components tested until the defect is found. The y-axis is scaled from 0 to 0.02 in increments of 0.005.\" \/><\/p>\n<p>The\u00a0[latex]y[\/latex]-axis contains the probability of [latex]x[\/latex], where [latex]X=[\/latex] the number of computer components tested.<\/p>\n<p>The number of components that you would expect to test until you find the first defective one is the mean,\u00a0[latex]\\displaystyle{\\mu}={50}[\/latex].<\/p>\n<p>The formula for the mean is\u00a0[latex]\\displaystyle{\\mu}=\\frac{{1}}{{p}}=\\frac{{1}}{{0.02}}={50}[\/latex]<\/p>\n<p>The formula for the variance is\u00a0[latex]\\displaystyle{\\sigma}^{{2}}={(\\frac{{1}}{{p}})}{(\\frac{{1}}{{p}}-{1})}={(\\frac{{1}}{{0.02}})}{(\\frac{{1}}{{0.02}}-{1})}={2},{450}[\/latex]<\/p>\n<p>The standard deviation is\u00a0[latex]\\displaystyle{\\sigma}=\\sqrt{{{(\\frac{{1}}{{p}})}{(\\frac{{1}}{{p}}-{1})}}}=\\sqrt{{{(\\frac{{1}}{{0.02}})}{(\\frac{{1}}{{0.02}}-{1})}}}={49.5}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The probability of a defective steel rod is 0.01. Steel rods are selected at random. Find the probability that the first defect occurs on the ninth steel rod. Use the TI-83+ or TI-84 calculator to find the answer.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q787284\">Show Solution<\/span><\/p>\n<div id=\"q787284\" class=\"hidden-answer\" style=\"display: none\">[latex]P(x=9)=0.0092[\/latex]<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The lifetime risk of developing pancreatic cancer is about one in 78 (1.28%). Let [latex]X=[\/latex]the number of people you ask until one says he or she has pancreatic cancer. Then [latex]X[\/latex] is a discrete random variable with a geometric distribution: [latex]\\displaystyle{X}~{G}{(\\frac{{1}}{{78}})}{\\quad\\text{or}\\quad}{X}~{G}{({0.0128})}[\/latex]<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>What is the probability of that you ask ten people before one says he or she has pancreatic cancer?<\/li>\n<li>What is the probability that you must ask 20 people?<\/li>\n<li>Find the (i) mean and (ii) standard deviation of [latex]X[\/latex].<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q556559\">Show Solution<\/span><\/p>\n<div id=\"q556559\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]P(x=10)=\\text{geometpdf}(0.0128,10)=0.0114[\/latex]<\/li>\n<li>[latex]P(x=20)=\\text{geometpdf}(0.0128,20)=0.01[\/latex]\n<ol style=\"list-style-type: lower-roman;\">\n<li>[latex]\\text{Mean}={\\mu}=\\frac{{1}}{{p}}=\\frac{{1}}{{0.0128}}={78}[\/latex]<\/li>\n<li>[latex]\\text{Standard Deviation}={\\sigma}=\\sqrt{{\\frac{{{1}-{p}}}{{{p}^{{2}}}}}}=\\sqrt{{\\frac{{{1}-{0.0128}}}{{0.0128}^{{2}}}}}\\approx{77.6234}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The literacy rate for a nation measures the proportion of people age 15 and over who can read and write. The literacy rate for women in Afghanistan is 12%. Let\u00a0[latex]X=[\/latex] the number of Afghani women you ask until one says that she is literate.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>What is the probability distribution of [latex]X[\/latex]?<\/li>\n<li>What is the probability that you ask five women before one says she is literate?<\/li>\n<li>What is the probability that you must ask ten women?<\/li>\n<li>Find the (i) mean and (ii) standard deviation of [latex]X[\/latex].<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q935609\">Show Solution<\/span><\/p>\n<div id=\"q935609\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]X{\\sim}G(0.12)[\/latex]<\/li>\n<li>[latex]P(x=5)=\\text{geometpdf}(0.12,5)=0.0720[\/latex]<\/li>\n<li>[latex]P(x=10)=\\text{geometpdf}(0.12,10)=0.0380[\/latex]\n<ol style=\"list-style-type: lower-roman;\">\n<li>[latex]\\text{Mean}={\\mu}=\\frac{{1}}{{p}}=\\frac{{1}}{{0.12}}\\approx{3333}[\/latex]<\/li>\n<li>[latex]\\text{Standard Deviation}={\\sigma}=\\sqrt{{\\frac{{{1}-{p}}}{{{p}^{{2}}}}}}=\\sqrt{{\\frac{{{1}-{0.12}}}{{{0.12}^{{2}}}}}}\\approx{7.8174}[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\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-1230\">\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>OpenStax, Statistics, Geometric Distribution. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/statistics\/pages\/4-4-geometric-distribution-optional\">https:\/\/openstax.org\/books\/statistics\/pages\/4-4-geometric-distribution-optional<\/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\/statistics\/pages\/1-introduction<\/li><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/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":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"OpenStax, Statistics, Geometric Distribution\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/statistics\/pages\/4-4-geometric-distribution-optional\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/statistics\/pages\/1-introduction\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"Open Stax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"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-1230","chapter","type-chapter","status-publish","hentry"],"part":240,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1230","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":6,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1230\/revisions"}],"predecessor-version":[{"id":3569,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1230\/revisions\/3569"}],"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\/1230\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1230"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1230"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1230"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1230"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}