{"id":3755,"date":"2023-10-20T17:49:59","date_gmt":"2023-10-20T17:49:59","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=3755"},"modified":"2025-03-25T14:23:49","modified_gmt":"2025-03-25T14:23:49","slug":"probability-using-venn-diagrams-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probability-using-venn-diagrams-learn-it-4\/","title":{"raw":"Probability with Venn Diagrams: Learn It 4","rendered":"Probability with Venn Diagrams: Learn It 4"},"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<section class=\"textbox keyTakeaway\">\r\n<h3>intersection of events<\/h3>\r\n<p>The intersection of two events, [latex]A[\/latex] and [latex]B[\/latex], is denoted by [latex]A \\cap B[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>[latex]A[\/latex] <strong>and<\/strong> [latex]B[\/latex] means that all elements in event [latex]A[\/latex] also belong in event [latex]B[\/latex] or equivalently, all elements of [latex]B[\/latex] is also in event [latex]A[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>In terms of probability, [latex]P(A \\text{ and }B)= P(A \\cap B)=[\/latex] the relative frequency of event [latex]A[\/latex] and [latex]B[\/latex] with respect to the sample space.<\/p>\r\n<\/section>\r\n<section>\r\n<section><\/section>\r\n<section class=\"textbox example\">\r\n<p>Consider the set [latex]A = \\{\\text{red}, \\text{blue}, \\text{green}\\}[\/latex] and [latex]B=\\{\\text{red},\\text{yellow},\\text{orange}\\}[\/latex].<br \/>\r\nThe event [latex]A \\cap B = \\{\\text{red}\\}[\/latex].<\/p>\r\n<\/section>\r\n<p>Using Venn diagrams, we can imagine AND events as the region where two events overlap. This is represented by the colored region in the Venn diagram below where two circles representing events [latex]A[\/latex] and [latex]B[\/latex] intersect.<\/p>\r\n<p>\u00a0 \u00a0 \u00a0<img class=\"size-medium wp-image-5983 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194643\/LI18.4a-300x207.png\" alt=\"This Venn diagram depicts two overlapping circles within a rectangle labeled &quot;S&quot;. The left circle is labeled &quot;A&quot;, the right circle is labeled &quot;B&quot;, and the overlapping area between the two circles is labeled &quot;A and B&quot;.\" width=\"300\" height=\"207\" \/><\/p>\r\n<section class=\"textbox example\">A survey of [latex]220[\/latex] random participants was conducted studying the behavior of adults age 18 \u2013 25 years old.\u00a0 According to the study, [latex]123[\/latex] of them were attendance college and [latex]153[\/latex] of them were working. Furthermore, [latex]56[\/latex] of them were both attending college and working.\r\n\r\n<p><strong>(a)<\/strong> Create a Venn diagram using the information given where C = \u201cthe event the participant attends college\u201d and W = \u201cthe event the participant is working\u201d.<\/p>\r\n<p>[reveal-answer q=\"583727\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"583727\"]<img class=\"alignnone size-medium wp-image-5982\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194641\/LI18.4d-300x207.png\" alt=\"In this Venn diagram, two overlapping circles are placed within a rectangle labeled &quot;S&quot;. The left circle is labeled &quot;C&quot; with the number &quot;67&quot; in the non-overlapping section. The right circle is labeled &quot;W&quot; with the number &quot;97&quot; in the non-overlapping section. The overlapping area between the two circles is labeled &quot;C and W&quot; and contains the number &quot;56&quot;.\" width=\"300\" height=\"207\" \/>[\/hidden-answer]<strong><br \/>\r\n(b)<\/strong> What is the probability that the participant is attending college and working?<\/p>\r\n<p>[reveal-answer q=\"740471\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"740471\"][latex]P(C \\cap W) = \\dfrac{56}{200} = \\dfrac{7}{25}[\/latex][\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]13087[\/ohm2_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>union of events<\/h3>\r\n<p>The union of two events, [latex]A[\/latex] or [latex]B[\/latex], is denoted by [latex]A \\cup B[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The union of two events consists of the set of all elements in the collection of both events. The outcomes in the event [latex]A[\/latex] or [latex]B[\/latex] are the outcomes that are in event [latex]A[\/latex], in event [latex]B[\/latex], or in both event [latex]A[\/latex] and event [latex]B[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>In term of probability, [latex]P(A \\text{ or }B)= P(A \\cup B)=[\/latex] the relative frequency of either event [latex]A[\/latex] or [latex]B[\/latex] (or both) with respect to the sample space.<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Consider the set [latex]A = \\{\\text{red}, \\text{blue}, \\text{green}\\}[\/latex] and [latex]B=\\{\\text{red},\\text{yellow},\\text{orange}\\}[\/latex].<\/p>\r\n<p>The event [latex]A \\cup B = \\{\\text{red}, \\text{blue}, \\text{green}, \\text{yellow}, \\text{orange}\\} [\/latex].<\/p>\r\n<\/section>\r\n<p>Once again, we can utilize the Venn diagram to represent this as two events [latex]A[\/latex] and [latex]B[\/latex] with both circles (and their intersection) being entirely shaded as shown below.<\/p>\r\n<p><img class=\"alignnone size-medium wp-image-5981 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194640\/LI18.4b-300x207.png\" alt=\"This Venn diagram contains two overlapping circles within a rectangle labeled &quot;S&quot;. The left circle is labeled &quot;A&quot; and the right circle is labeled &quot;B&quot;. The region inside all of the circles is shaded to represent the Union.\" width=\"300\" height=\"207\" \/><\/p>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]13088[\/ohm2_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>mutually exclusive events<\/h3>\r\n<p><strong>Mutually exclusive<\/strong> describes two or more events that cannot happen simultaneously.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>If event [latex]A[\/latex] and event [latex]B[\/latex] are mutually exclusive, then [latex]A \\cap B[\/latex] is an empty set and [latex]P(A \\cap B) = 0[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>Therefore,<\/p>\r\n<p style=\"text-align: center;\">[latex]P(A \\text{ or } B) = P(A \\cup B) = P(A)+P(B)[\/latex]<\/p>\r\n<\/section>\r\n<p>Using Venn diagrams, we can also visualize the concept of mutually exclusive or disjoint events as non-intersecting circles as shown below.<\/p>\r\n<p><img class=\"alignnone size-medium wp-image-5979 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194637\/LI18.4e-300x207.png\" alt=\"This diagram presents two non-overlapping circles within a rectangle labeled &quot;S&quot;. The left circle is labeled &quot;A&quot;, and the right circle is labeled &quot;B&quot;.\" width=\"300\" height=\"207\" \/><\/p>\r\n<p>When we find probability [latex]P(A \\text{ or } B) [\/latex] for mutually exclusive events, we would consider the shaded region of both circles without any intersection.<\/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<section class=\"textbox keyTakeaway\">\n<h3>intersection of events<\/h3>\n<p>The intersection of two events, [latex]A[\/latex] and [latex]B[\/latex], is denoted by [latex]A \\cap B[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>[latex]A[\/latex] <strong>and<\/strong> [latex]B[\/latex] means that all elements in event [latex]A[\/latex] also belong in event [latex]B[\/latex] or equivalently, all elements of [latex]B[\/latex] is also in event [latex]A[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>In terms of probability, [latex]P(A \\text{ and }B)= P(A \\cap B)=[\/latex] the relative frequency of event [latex]A[\/latex] and [latex]B[\/latex] with respect to the sample space.<\/p>\n<\/section>\n<section>\n<section><\/section>\n<section class=\"textbox example\">\n<p>Consider the set [latex]A = \\{\\text{red}, \\text{blue}, \\text{green}\\}[\/latex] and [latex]B=\\{\\text{red},\\text{yellow},\\text{orange}\\}[\/latex].<br \/>\nThe event [latex]A \\cap B = \\{\\text{red}\\}[\/latex].<\/p>\n<\/section>\n<p>Using Venn diagrams, we can imagine AND events as the region where two events overlap. This is represented by the colored region in the Venn diagram below where two circles representing events [latex]A[\/latex] and [latex]B[\/latex] intersect.<\/p>\n<p>\u00a0 \u00a0 \u00a0<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-5983 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194643\/LI18.4a-300x207.png\" alt=\"This Venn diagram depicts two overlapping circles within a rectangle labeled &quot;S&quot;. The left circle is labeled &quot;A&quot;, the right circle is labeled &quot;B&quot;, and the overlapping area between the two circles is labeled &quot;A and B&quot;.\" width=\"300\" height=\"207\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194643\/LI18.4a-300x207.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194643\/LI18.4a-65x45.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194643\/LI18.4a-225x155.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194643\/LI18.4a-350x241.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194643\/LI18.4a.png 581w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<section class=\"textbox example\">A survey of [latex]220[\/latex] random participants was conducted studying the behavior of adults age 18 \u2013 25 years old.\u00a0 According to the study, [latex]123[\/latex] of them were attendance college and [latex]153[\/latex] of them were working. Furthermore, [latex]56[\/latex] of them were both attending college and working.<\/p>\n<p><strong>(a)<\/strong> Create a Venn diagram using the information given where C = \u201cthe event the participant attends college\u201d and W = \u201cthe event the participant is working\u201d.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q583727\">Show Answer<\/button><\/p>\n<div id=\"q583727\" class=\"hidden-answer\" style=\"display: none\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5982\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194641\/LI18.4d-300x207.png\" alt=\"In this Venn diagram, two overlapping circles are placed within a rectangle labeled &quot;S&quot;. The left circle is labeled &quot;C&quot; with the number &quot;67&quot; in the non-overlapping section. The right circle is labeled &quot;W&quot; with the number &quot;97&quot; in the non-overlapping section. The overlapping area between the two circles is labeled &quot;C and W&quot; and contains the number &quot;56&quot;.\" width=\"300\" height=\"207\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194641\/LI18.4d-300x207.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194641\/LI18.4d-65x45.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194641\/LI18.4d-225x155.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194641\/LI18.4d-350x241.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194641\/LI18.4d.png 581w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/div>\n<\/div>\n<p><strong><br \/>\n(b)<\/strong> What is the probability that the participant is attending college and working?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q740471\">Show Answer<\/button><\/p>\n<div id=\"q740471\" class=\"hidden-answer\" style=\"display: none\">[latex]P(C \\cap W) = \\dfrac{56}{200} = \\dfrac{7}{25}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13087\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13087&theme=lumen&iframe_resize_id=ohm13087&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>union of events<\/h3>\n<p>The union of two events, [latex]A[\/latex] or [latex]B[\/latex], is denoted by [latex]A \\cup B[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>The union of two events consists of the set of all elements in the collection of both events. The outcomes in the event [latex]A[\/latex] or [latex]B[\/latex] are the outcomes that are in event [latex]A[\/latex], in event [latex]B[\/latex], or in both event [latex]A[\/latex] and event [latex]B[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>In term of probability, [latex]P(A \\text{ or }B)= P(A \\cup B)=[\/latex] the relative frequency of either event [latex]A[\/latex] or [latex]B[\/latex] (or both) with respect to the sample space.<\/p>\n<\/section>\n<section class=\"textbox example\">\n<p>Consider the set [latex]A = \\{\\text{red}, \\text{blue}, \\text{green}\\}[\/latex] and [latex]B=\\{\\text{red},\\text{yellow},\\text{orange}\\}[\/latex].<\/p>\n<p>The event [latex]A \\cup B = \\{\\text{red}, \\text{blue}, \\text{green}, \\text{yellow}, \\text{orange}\\}[\/latex].<\/p>\n<\/section>\n<p>Once again, we can utilize the Venn diagram to represent this as two events [latex]A[\/latex] and [latex]B[\/latex] with both circles (and their intersection) being entirely shaded as shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5981 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194640\/LI18.4b-300x207.png\" alt=\"This Venn diagram contains two overlapping circles within a rectangle labeled &quot;S&quot;. The left circle is labeled &quot;A&quot; and the right circle is labeled &quot;B&quot;. The region inside all of the circles is shaded to represent the Union.\" width=\"300\" height=\"207\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194640\/LI18.4b-300x207.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194640\/LI18.4b-65x45.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194640\/LI18.4b-225x155.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194640\/LI18.4b-350x241.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194640\/LI18.4b.png 581w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13088\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13088&theme=lumen&iframe_resize_id=ohm13088&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>mutually exclusive events<\/h3>\n<p><strong>Mutually exclusive<\/strong> describes two or more events that cannot happen simultaneously.<\/p>\n<p>&nbsp;<\/p>\n<p>If event [latex]A[\/latex] and event [latex]B[\/latex] are mutually exclusive, then [latex]A \\cap B[\/latex] is an empty set and [latex]P(A \\cap B) = 0[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>Therefore,<\/p>\n<p style=\"text-align: center;\">[latex]P(A \\text{ or } B) = P(A \\cup B) = P(A)+P(B)[\/latex]<\/p>\n<\/section>\n<p>Using Venn diagrams, we can also visualize the concept of mutually exclusive or disjoint events as non-intersecting circles as shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5979 aligncenter\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194637\/LI18.4e-300x207.png\" alt=\"This diagram presents two non-overlapping circles within a rectangle labeled &quot;S&quot;. The left circle is labeled &quot;A&quot;, and the right circle is labeled &quot;B&quot;.\" width=\"300\" height=\"207\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194637\/LI18.4e-300x207.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194637\/LI18.4e-65x45.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194637\/LI18.4e-225x155.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194637\/LI18.4e-350x241.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/10\/26194637\/LI18.4e.png 581w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>When we find probability [latex]P(A \\text{ or } B)[\/latex] for mutually exclusive events, we would consider the shaded region of both circles without any intersection.<\/p>\n<\/section>\n","protected":false},"author":12,"menu_order":7,"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\/3755"}],"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\/12"}],"version-history":[{"count":25,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3755\/revisions"}],"predecessor-version":[{"id":6353,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3755\/revisions\/6353"}],"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\/3755\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=3755"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=3755"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=3755"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=3755"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}