{"id":3306,"date":"2023-10-02T05:34:39","date_gmt":"2023-10-02T05:34:39","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=3306"},"modified":"2025-03-25T14:23:51","modified_gmt":"2025-03-25T14:23:51","slug":"probability-using-venn-diagrams-learn-it-3-update","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probability-using-venn-diagrams-learn-it-3-update\/","title":{"raw":"Probability with Venn Diagrams: Learn It 5","rendered":"Probability with Venn Diagrams: Learn It 5"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand the concept of set theory and how it relates to probability<\/li>\r\n\t<li>Create and interpret Venn diagrams to visually represent sets and their intersections<\/li>\r\n\t<li>Understand how to use Venn diagrams to solve problems related to probability, including union, intersection, and complement of events<\/li>\r\n<\/ul>\r\n<\/section>\r\n<p>Venn diagrams can also be useful when trying to visualize conditional probability.<\/p>\r\n<p><strong>Conditional probability<\/strong> is a way to calculate the likelihood of an event happening, considering that another event has already occurred. It's like adjusting the probability based on additional information you have.<\/p>\r\n<section class=\"textbox recall\">\r\n<p>The <strong>conditional probability\u00a0<\/strong>of [latex]A[\/latex]<span style=\"background-color: initial; font-size: 0.9em;\">\u00a0<\/span>given [latex]B[\/latex], denoted as [latex]P(A\\text{ given }B) = P(A|B)[\/latex], represents the represents the probability of event [latex]A[\/latex] occurring given that event [latex]B[\/latex] has occurred.\u00a0<\/p>\r\n<p>The formula is given by:<\/p>\r\n<p style=\"text-align: center;\">[latex]P(A|B) = \\dfrac{\\text{Number of times both A and B occur}}{\\text{Number of times B occurs}}[\/latex]<\/p>\r\n<p>OR<\/p>\r\n<p style=\"text-align: center;\">[latex]P(A|B) = \\dfrac{P(A \\cap B)}{P(B)}[\/latex]<\/p>\r\n<p>Where:<\/p>\r\n<ul>\r\n\t<li>[latex]P(A|B)[\/latex] is the condition probability of event [latex]A[\/latex] occurring given event [latex]B[\/latex] has occurred.<\/li>\r\n\t<li>[latex]P(A\\text{ and }B)=P(A \\cap B)[\/latex] is the probability of both events occurring<\/li>\r\n\t<li>[latex]P(B)[\/latex] is the probability of event [latex]B[\/latex] occurring.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p><span style=\"font-size: 16.8px;\">Suppose in a study was conducted on pet ownership. Out of the [latex]50[\/latex] participants, the study discovered that there were [latex]40\\%[\/latex] cat owners and [latex]60\\%[\/latex] were dog owners. Furthermore, [latex]10\\%[\/latex] of the participants were both cat and dog owners. <\/span><\/p>\r\n<p><span style=\"font-size: 16.8px;\">Let [latex]C[\/latex] be the event someone is a cat owner and [latex]D[\/latex] be the event someone is a dog owner.<br \/>\r\n<\/span><\/p>\r\n<p><strong>(a)<\/strong> Construct the Venn diagram for the given scenario with the counts for the different events.<br \/>\r\n[reveal-answer q=\"625458\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"625458\"]<img class=\"alignnone size-medium wp-image-5990\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/28171308\/LI18.5b-300x207.png\" alt=\"This Venn diagram showcases two overlapping circles within a rectangle labeled &quot;S&quot;. The left circle is labeled &quot;C&quot; and contains the number &quot;15&quot; in the non-overlapping section. The right circle is labeled &quot;D&quot; and contains the number &quot;25&quot; in its non-overlapping section. The overlapping area between the two circles contains the number &quot;5&quot;.\" width=\"300\" height=\"207\" \/><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[\/hidden-answer]<\/span><\/p>\r\n<p><strong>(b)<\/strong> How many people are cat owners?<br \/>\r\n[reveal-answer q=\"63790\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"63790\"][latex]20[\/latex][\/hidden-answer]<\/p>\r\n<p><strong>(c)\u00a0<\/strong>How many people are in this study are cat or dog owners?<br \/>\r\n[reveal-answer q=\"880492\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"880492\"][latex]15+5+25 = 45[\/latex][\/hidden-answer]<\/p>\r\n<p><strong>(d)\u00a0<\/strong>How many cat owners own a dog?<br \/>\r\n[reveal-answer q=\"713795\"]Show Answer[\/reveal-answer][hidden-answer a=\"713795\"]To find out how many cat owners also own a dog, we need to consider the overlapping portion of cat and dog owners, which corresponds to [latex]5[\/latex] participants who own both.[\/hidden-answer]\u00a0<\/p>\r\n<p><strong>(e) <\/strong>Given that a person is cat owner, what is the probability they are also a dog owner?<\/p>\r\n<p>[reveal-answer q=\"249861\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"249861\"]This is a conditional probability problem!\u00a0<\/p>\r\n<p>[latex]P(D \\text{ given } C) = \\dfrac{\\text{Number of times both D and C occur}}{\\text{Number of times C occur}} = \\dfrac{\\text{Number of cat owners who also own dogs}}{\\text{Number of cat owners}}=\\dfrac{5}{20}[\/latex][\/hidden-answer]<\/p>\r\n<\/section>\r\n<section><\/section>\r\n<section><\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]13784[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand the concept of set theory and how it relates to probability<\/li>\n<li>Create and interpret Venn diagrams to visually represent sets and their intersections<\/li>\n<li>Understand how to use Venn diagrams to solve problems related to probability, including union, intersection, and complement of events<\/li>\n<\/ul>\n<\/section>\n<p>Venn diagrams can also be useful when trying to visualize conditional probability.<\/p>\n<p><strong>Conditional probability<\/strong> is a way to calculate the likelihood of an event happening, considering that another event has already occurred. It&#8217;s like adjusting the probability based on additional information you have.<\/p>\n<section class=\"textbox recall\">\n<p>The <strong>conditional probability\u00a0<\/strong>of [latex]A[\/latex]<span style=\"background-color: initial; font-size: 0.9em;\">\u00a0<\/span>given [latex]B[\/latex], denoted as [latex]P(A\\text{ given }B) = P(A|B)[\/latex], represents the represents the probability of event [latex]A[\/latex] occurring given that event [latex]B[\/latex] has occurred.\u00a0<\/p>\n<p>The formula is given by:<\/p>\n<p style=\"text-align: center;\">[latex]P(A|B) = \\dfrac{\\text{Number of times both A and B occur}}{\\text{Number of times B occurs}}[\/latex]<\/p>\n<p>OR<\/p>\n<p style=\"text-align: center;\">[latex]P(A|B) = \\dfrac{P(A \\cap B)}{P(B)}[\/latex]<\/p>\n<p>Where:<\/p>\n<ul>\n<li>[latex]P(A|B)[\/latex] is the condition probability of event [latex]A[\/latex] occurring given event [latex]B[\/latex] has occurred.<\/li>\n<li>[latex]P(A\\text{ and }B)=P(A \\cap B)[\/latex] is the probability of both events occurring<\/li>\n<li>[latex]P(B)[\/latex] is the probability of event [latex]B[\/latex] occurring.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox example\">\n<p><span style=\"font-size: 16.8px;\">Suppose in a study was conducted on pet ownership. Out of the [latex]50[\/latex] participants, the study discovered that there were [latex]40\\%[\/latex] cat owners and [latex]60\\%[\/latex] were dog owners. Furthermore, [latex]10\\%[\/latex] of the participants were both cat and dog owners. <\/span><\/p>\n<p><span style=\"font-size: 16.8px;\">Let [latex]C[\/latex] be the event someone is a cat owner and [latex]D[\/latex] be the event someone is a dog owner.<br \/>\n<\/span><\/p>\n<p><strong>(a)<\/strong> Construct the Venn diagram for the given scenario with the counts for the different events.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q625458\">Show Answer<\/button><\/p>\n<div id=\"q625458\" class=\"hidden-answer\" style=\"display: none\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5990\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/28171308\/LI18.5b-300x207.png\" alt=\"This Venn diagram showcases two overlapping circles within a rectangle labeled &quot;S&quot;. The left circle is labeled &quot;C&quot; and contains the number &quot;15&quot; in the non-overlapping section. The right circle is labeled &quot;D&quot; and contains the number &quot;25&quot; in its non-overlapping section. The overlapping area between the two circles contains the number &quot;5&quot;.\" width=\"300\" height=\"207\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/28171308\/LI18.5b-300x207.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/28171308\/LI18.5b-65x45.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/28171308\/LI18.5b-225x155.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/28171308\/LI18.5b-350x241.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/28171308\/LI18.5b.png 581w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<p><strong>(b)<\/strong> How many people are cat owners?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q63790\">Show Answer<\/button><\/p>\n<div id=\"q63790\" class=\"hidden-answer\" style=\"display: none\">[latex]20[\/latex]<\/div>\n<\/div>\n<p><strong>(c)\u00a0<\/strong>How many people are in this study are cat or dog owners?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q880492\">Show Answer<\/button><\/p>\n<div id=\"q880492\" class=\"hidden-answer\" style=\"display: none\">[latex]15+5+25 = 45[\/latex]<\/div>\n<\/div>\n<p><strong>(d)\u00a0<\/strong>How many cat owners own a dog?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q713795\">Show Answer<\/button><\/p>\n<div id=\"q713795\" class=\"hidden-answer\" style=\"display: none\">To find out how many cat owners also own a dog, we need to consider the overlapping portion of cat and dog owners, which corresponds to [latex]5[\/latex] participants who own both.<\/div>\n<\/div>\n<p>\u00a0<\/p>\n<p><strong>(e) <\/strong>Given that a person is cat owner, what is the probability they are also a dog owner?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q249861\">Show Answer<\/button><\/p>\n<div id=\"q249861\" class=\"hidden-answer\" style=\"display: none\">This is a conditional probability problem!\u00a0<\/p>\n<p>[latex]P(D \\text{ given } C) = \\dfrac{\\text{Number of times both D and C occur}}{\\text{Number of times C occur}} = \\dfrac{\\text{Number of cat owners who also own dogs}}{\\text{Number of cat owners}}=\\dfrac{5}{20}[\/latex]<\/p><\/div>\n<\/div>\n<\/section>\n<section><\/section>\n<section><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm13784\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13784&theme=lumen&iframe_resize_id=ohm13784&source=tnh\" width=\"100%\" 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