{"id":402,"date":"2023-06-05T15:31:03","date_gmt":"2023-06-05T15:31:03","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/chapter\/conditional-probability\/"},"modified":"2025-12-08T00:23:25","modified_gmt":"2025-12-08T00:23:25","slug":"conditional-probability","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/chapter\/conditional-probability\/","title":{"raw":"Conditional Probability","rendered":"Conditional Probability"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Calculate a conditional probability using standard notation<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>LEARNING PROBABILISTIC PROCESSES<\/h3>\r\nRemember to work through each example in the text and in the EXAMPLE and TRY IT boxes with a pencil on paper, pausing as frequently as needed to digest the process. Watch the videos by working them out on paper, pausing the video as frequently as you need to make sense of the demonstration. Don't be afraid to ask for help -- hard work and willingness to learn translate into success!\r\n\r\n<\/div>\r\nIn this section, we will consider events where additional information is given. <span class=\"T286Pc\" data-sfc-cp=\"\">Conditions on an event means you are considering a new, smaller set of outcomes where that event has already occurred. The probability of such events is <\/span>called <strong>conditional probabilities<\/strong>.\r\n<div class=\"textbox\">\r\n<h3>Conditional Probability<\/h3>\r\nThe probability the event <em>B<\/em> occurs, given that event <em>A<\/em> has happened, is represented as\r\n<p style=\"text-align: center;\"><em>P<\/em>(<em>B<\/em> | <em>A<\/em>)<\/p>\r\nThis is read as \u201cthe probability of <em>B<\/em> given <em>A<\/em>\u201d\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe table below shows the number of survey subjects who have received and not received a speeding ticket in the last year, and the color of their car. Find the probability that a randomly chosen person:\r\n<ol>\r\n \t<li>has a speeding ticket <em>given<\/em> they have a red car<\/li>\r\n \t<li>has a red car <em>given<\/em> they have a speeding ticket<\/li>\r\n<\/ol>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Speeding ticket<\/td>\r\n<td>No speeding ticket<\/td>\r\n<td>Total<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Red car<\/td>\r\n<td>15<\/td>\r\n<td>135<\/td>\r\n<td>150<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Not red car<\/td>\r\n<td>45<\/td>\r\n<td>470<\/td>\r\n<td>515<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Total<\/td>\r\n<td>60<\/td>\r\n<td>605<\/td>\r\n<td>665<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"849340\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"849340\"]\r\n<ol>\r\n \t<li>Since we know the person has a red car, we are only considering the 150 people in the first row of the table. Of those, 15 have a speeding ticket, so P(ticket | red car) = [latex]\\frac{15}{150}=\\frac{1}{10}=0.1[\/latex]<\/li>\r\n \t<li>Since we know the person has a speeding ticket, we are only considering the 60 people in the first column of the table. Of those, 15 have a red car, so P(red car | ticket) = [latex]\\frac{15}{60}=\\frac{1}{4}=0.25[\/latex].<\/li>\r\n<\/ol>\r\nNotice from the last example that P(B | A) is <strong>not<\/strong> equal to P(A | B).\r\n\r\n[\/hidden-answer]\r\n\r\nThese kinds of conditional probabilities are what insurance companies use to determine your insurance rates. They look at the conditional probability of you having accident, given your age, your car, your car color, your driving history, etc., and price your policy based on that likelihood.\r\n\r\nView more about conditional probability in the following video.\r\n\r\nhttps:\/\/youtu.be\/b6tstekMlb8\r\n\r\n<\/div>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA home pregnancy test was given to women, then pregnancy was verified through blood tests. \u00a0The following table shows the home pregnancy test results.\r\n\r\nFind\r\n<ol>\r\n \t<li><em>P<\/em>(not pregnant | positive test result)<\/li>\r\n \t<li><em>P<\/em>(positive test result | not pregnant)<\/li>\r\n<\/ol>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><\/td>\r\n<td>Positive test<\/td>\r\n<td>Negative test<\/td>\r\n<td>Total<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Pregnant<\/td>\r\n<td>70<\/td>\r\n<td>4<\/td>\r\n<td>74<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Not Pregnant<\/td>\r\n<td>5<\/td>\r\n<td>14<\/td>\r\n<td>19<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Total<\/td>\r\n<td>75<\/td>\r\n<td>18<\/td>\r\n<td>93<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"968710\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"968710\"]\r\n<ol>\r\n \t<li>Since we know the test result was positive, we\u2019re limited to the 75 women in the first column, of which 5 were not pregnant. <em>P<\/em>(not pregnant | positive test result) = [latex]\\frac{5}{75}\\approx0.067[\/latex].<\/li>\r\n \t<li>Since we know the woman is not pregnant, we are limited to the 19 women in the second row, of which 5 had a positive test.\u00a0<em>P<\/em>(positive test result | not pregnant) = [latex]\\frac{5}{19}\\approx0.263[\/latex]<\/li>\r\n<\/ol>\r\nThe second result is what is usually called a false positive: A positive result when the woman is not actually pregnant.\r\n\r\n[\/hidden-answer]\r\n\r\nSee more about this example here.\r\n\r\nhttps:\/\/youtu.be\/LH0cuHS9Ez0\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]7116[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Calculate a conditional probability using standard notation<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<h3>LEARNING PROBABILISTIC PROCESSES<\/h3>\n<p>Remember to work through each example in the text and in the EXAMPLE and TRY IT boxes with a pencil on paper, pausing as frequently as needed to digest the process. Watch the videos by working them out on paper, pausing the video as frequently as you need to make sense of the demonstration. Don&#8217;t be afraid to ask for help &#8212; hard work and willingness to learn translate into success!<\/p>\n<\/div>\n<p>In this section, we will consider events where additional information is given. <span class=\"T286Pc\" data-sfc-cp=\"\">Conditions on an event means you are considering a new, smaller set of outcomes where that event has already occurred. The probability of such events is <\/span>called <strong>conditional probabilities<\/strong>.<\/p>\n<div class=\"textbox\">\n<h3>Conditional Probability<\/h3>\n<p>The probability the event <em>B<\/em> occurs, given that event <em>A<\/em> has happened, is represented as<\/p>\n<p style=\"text-align: center;\"><em>P<\/em>(<em>B<\/em> | <em>A<\/em>)<\/p>\n<p>This is read as \u201cthe probability of <em>B<\/em> given <em>A<\/em>\u201d<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The table below shows the number of survey subjects who have received and not received a speeding ticket in the last year, and the color of their car. Find the probability that a randomly chosen person:<\/p>\n<ol>\n<li>has a speeding ticket <em>given<\/em> they have a red car<\/li>\n<li>has a red car <em>given<\/em> they have a speeding ticket<\/li>\n<\/ol>\n<table>\n<tbody>\n<tr>\n<td>Speeding ticket<\/td>\n<td>No speeding ticket<\/td>\n<td>Total<\/td>\n<\/tr>\n<tr>\n<td>Red car<\/td>\n<td>15<\/td>\n<td>135<\/td>\n<td>150<\/td>\n<\/tr>\n<tr>\n<td>Not red car<\/td>\n<td>45<\/td>\n<td>470<\/td>\n<td>515<\/td>\n<\/tr>\n<tr>\n<td>Total<\/td>\n<td>60<\/td>\n<td>605<\/td>\n<td>665<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q849340\">Show Solution<\/span><\/p>\n<div id=\"q849340\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Since we know the person has a red car, we are only considering the 150 people in the first row of the table. Of those, 15 have a speeding ticket, so P(ticket | red car) = [latex]\\frac{15}{150}=\\frac{1}{10}=0.1[\/latex]<\/li>\n<li>Since we know the person has a speeding ticket, we are only considering the 60 people in the first column of the table. Of those, 15 have a red car, so P(red car | ticket) = [latex]\\frac{15}{60}=\\frac{1}{4}=0.25[\/latex].<\/li>\n<\/ol>\n<p>Notice from the last example that P(B | A) is <strong>not<\/strong> equal to P(A | B).<\/p>\n<\/div>\n<\/div>\n<p>These kinds of conditional probabilities are what insurance companies use to determine your insurance rates. They look at the conditional probability of you having accident, given your age, your car, your car color, your driving history, etc., and price your policy based on that likelihood.<\/p>\n<p>View more about conditional probability in the following video.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Basic conditional probability\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/b6tstekMlb8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A home pregnancy test was given to women, then pregnancy was verified through blood tests. \u00a0The following table shows the home pregnancy test results.<\/p>\n<p>Find<\/p>\n<ol>\n<li><em>P<\/em>(not pregnant | positive test result)<\/li>\n<li><em>P<\/em>(positive test result | not pregnant)<\/li>\n<\/ol>\n<table>\n<tbody>\n<tr>\n<td><\/td>\n<td>Positive test<\/td>\n<td>Negative test<\/td>\n<td>Total<\/td>\n<\/tr>\n<tr>\n<td>Pregnant<\/td>\n<td>70<\/td>\n<td>4<\/td>\n<td>74<\/td>\n<\/tr>\n<tr>\n<td>Not Pregnant<\/td>\n<td>5<\/td>\n<td>14<\/td>\n<td>19<\/td>\n<\/tr>\n<tr>\n<td>Total<\/td>\n<td>75<\/td>\n<td>18<\/td>\n<td>93<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q968710\">Show Solution<\/span><\/p>\n<div id=\"q968710\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Since we know the test result was positive, we\u2019re limited to the 75 women in the first column, of which 5 were not pregnant. <em>P<\/em>(not pregnant | positive test result) = [latex]\\frac{5}{75}\\approx0.067[\/latex].<\/li>\n<li>Since we know the woman is not pregnant, we are limited to the 19 women in the second row, of which 5 had a positive test.\u00a0<em>P<\/em>(positive test result | not pregnant) = [latex]\\frac{5}{19}\\approx0.263[\/latex]<\/li>\n<\/ol>\n<p>The second result is what is usually called a false positive: A positive result when the woman is not actually pregnant.<\/p>\n<\/div>\n<\/div>\n<p>See more about this example here.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Conditional probability from a table\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/LH0cuHS9Ez0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm7116\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=7116&theme=oea&iframe_resize_id=ohm7116&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/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-402\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Math in Society. <strong>Authored by<\/strong>: Lippman, David. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Statistics, Describing Data, and Probability . <strong>Authored by<\/strong>: Jeff Eldridge. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/\">CC BY-SA: Attribution-ShareAlike<\/a><\/em><\/li><li>Question ID 7118. <strong>Authored by<\/strong>: WebWork-Rochester. <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>: IMathAS Community License CC-BY + GPL<\/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":503070,"menu_order":19,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"Lippman, 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