{"id":3160,"date":"2023-09-14T21:55:07","date_gmt":"2023-09-14T21:55:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=3160"},"modified":"2025-05-17T03:00:52","modified_gmt":"2025-05-17T03:00:52","slug":"probability-of-two-events-learn-it-3-update","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probability-of-two-events-learn-it-3-update\/","title":{"raw":"Probability with Venn Diagrams: Learn It 3","rendered":"Probability with Venn Diagrams: Learn It 3"},"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>One of the advantages of Venn diagrams is their ability to visualize the probability of events. This is particularly helpful when we want to visualize the probability rules of <strong>AND, OR,<\/strong> and\u00a0<strong>NOT<\/strong>.<\/p>\r\n<p>Let's start with <strong>NOT<\/strong>.<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>complement of an event<\/h3>\r\n<p>The <strong>complement <\/strong>of event [latex]A[\/latex] is denoted [latex]A'[\/latex] (read \"A prime\") or [latex]A^c[\/latex] (read \"A complement\").<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The complement of an event [latex]A[\/latex] consist of all outcomes that are\u00a0<strong>NOT<\/strong> in [latex]A[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>More generally, for any event [latex]A[\/latex], we can think of the probability of complements as having the following relationship:<\/p>\r\n<p style=\"text-align: center;\">[latex]P[\/latex]([latex]A[\/latex]) + [latex]P[\/latex](not [latex]A[\/latex]) = [latex]1[\/latex]<\/p>\r\n<p style=\"text-align: center;\">or<\/p>\r\n<p style=\"text-align: center;\">[latex]P[\/latex]([latex]A[\/latex]) + [latex]P[\/latex]([latex]A'[\/latex]) = [latex]1[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Suppose the sample space is [latex]S =[\/latex] all whole numbers from [latex]1[\/latex] to [latex]9[\/latex].<\/p>\r\n<p>If [latex]A = \\{1,2,4\\}[\/latex], then [latex]A' = \\{3,5,6,7,8,9\\}[\/latex].<\/p>\r\n<\/section>\r\n<p>Using the Venn diagram, we can represent the complement of the event [latex]A[\/latex] as the shaded region outside the circle representing the event. It's important to note that these two regions together represent the entire sample space.<\/p>\r\n<p style=\"text-align: center;\"><strong><img class=\"alignnone size-medium wp-image-5973\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26192827\/LI18.3-300x210.png\" alt=\"A rectangle labeled S with a circle labeled A inside it.\" width=\"300\" height=\"210\" \/><\/strong><\/p>\r\n<section class=\"textbox example\">According to Census.gov, in 2022 only 37% of adults 25 years or older in the United States had at least a bachelor\u2019s degree.\r\n\r\n<p><strong>(a)<\/strong> Suppose we had a sample of 100 adults, how would we create a Venn diagram representing the event B = \u201chas a bachelor\u2019s degree\u201d and its complement using the numbers given?<br \/>\r\n<br \/>\r\n[reveal-answer q=\"970823\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"970823\"]<img class=\"alignnone size-medium wp-image-5986\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/27170407\/LI18.3b-300x210.png\" alt=\"This Venn diagram is composed of a single circle inside a rectangle labeled &quot;S&quot;. The circle is labeled &quot;B&quot; and contains the number &quot;37&quot;. Outside of the circle, but still within the rectangle, there is a number &quot;63&quot;.\" width=\"300\" height=\"210\" \/>[\/hidden-answer]<\/p>\r\n<p><strong>(b)<\/strong> What is the probability that an adult 25 years or older in the United States does not have a bachelor\u2019s degree?\u00a0<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"391869\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"391869\"][latex]P(B') = 1 - \\frac{37}{100} = \\frac{63}{100}[\/latex][\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]13085[\/ohm2_question]<\/p>\r\n<\/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>One of the advantages of Venn diagrams is their ability to visualize the probability of events. This is particularly helpful when we want to visualize the probability rules of <strong>AND, OR,<\/strong> and\u00a0<strong>NOT<\/strong>.<\/p>\n<p>Let&#8217;s start with <strong>NOT<\/strong>.<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>complement of an event<\/h3>\n<p>The <strong>complement <\/strong>of event [latex]A[\/latex] is denoted [latex]A'[\/latex] (read &#8220;A prime&#8221;) or [latex]A^c[\/latex] (read &#8220;A complement&#8221;).<\/p>\n<p>&nbsp;<\/p>\n<p>The complement of an event [latex]A[\/latex] consist of all outcomes that are\u00a0<strong>NOT<\/strong> in [latex]A[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>More generally, for any event [latex]A[\/latex], we can think of the probability of complements as having the following relationship:<\/p>\n<p style=\"text-align: center;\">[latex]P[\/latex]([latex]A[\/latex]) + [latex]P[\/latex](not [latex]A[\/latex]) = [latex]1[\/latex]<\/p>\n<p style=\"text-align: center;\">or<\/p>\n<p style=\"text-align: center;\">[latex]P[\/latex]([latex]A[\/latex]) + [latex]P[\/latex]([latex]A'[\/latex]) = [latex]1[\/latex]<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Suppose the sample space is [latex]S =[\/latex] all whole numbers from [latex]1[\/latex] to [latex]9[\/latex].<\/p>\n<p>If [latex]A = \\{1,2,4\\}[\/latex], then [latex]A' = \\{3,5,6,7,8,9\\}[\/latex].<\/p>\n<\/section>\n<p>Using the Venn diagram, we can represent the complement of the event [latex]A[\/latex] as the shaded region outside the circle representing the event. It&#8217;s important to note that these two regions together represent the entire sample space.<\/p>\n<p style=\"text-align: center;\"><strong><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5973\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26192827\/LI18.3-300x210.png\" alt=\"A rectangle labeled S with a circle labeled A inside it.\" width=\"300\" height=\"210\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26192827\/LI18.3-300x210.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26192827\/LI18.3-65x46.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26192827\/LI18.3-225x158.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26192827\/LI18.3-350x245.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26192827\/LI18.3.png 500w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/strong><\/p>\n<section class=\"textbox example\">According to Census.gov, in 2022 only 37% of adults 25 years or older in the United States had at least a bachelor\u2019s degree.<\/p>\n<p><strong>(a)<\/strong> Suppose we had a sample of 100 adults, how would we create a Venn diagram representing the event B = \u201chas a bachelor\u2019s degree\u201d and its complement using the numbers given?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q970823\">Show Answer<\/button><\/p>\n<div id=\"q970823\" class=\"hidden-answer\" style=\"display: none\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5986\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/27170407\/LI18.3b-300x210.png\" alt=\"This Venn diagram is composed of a single circle inside a rectangle labeled &quot;S&quot;. The circle is labeled &quot;B&quot; and contains the number &quot;37&quot;. Outside of the circle, but still within the rectangle, there is a number &quot;63&quot;.\" width=\"300\" height=\"210\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/27170407\/LI18.3b-300x210.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/27170407\/LI18.3b-65x46.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/27170407\/LI18.3b-225x158.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/27170407\/LI18.3b-350x245.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/27170407\/LI18.3b.png 500w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/div>\n<\/div>\n<p><strong>(b)<\/strong> What is the probability that an adult 25 years or older in the United States does not have a bachelor\u2019s degree?\u00a0<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q391869\">Show Answer<\/button><\/p>\n<div id=\"q391869\" class=\"hidden-answer\" style=\"display: none\">[latex]P(B') = 1 - \\frac{37}{100} = \\frac{63}{100}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13085\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13085&theme=lumen&iframe_resize_id=ohm13085&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":51,"menu_order":6,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":2910,"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\/3160"}],"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\/51"}],"version-history":[{"count":30,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3160\/revisions"}],"predecessor-version":[{"id":6939,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3160\/revisions\/6939"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/2910"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3160\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=3160"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=3160"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=3160"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=3160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}