{"id":3234,"date":"2023-09-25T05:14:26","date_gmt":"2023-09-25T05:14:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=3234"},"modified":"2025-06-02T19:29:56","modified_gmt":"2025-06-02T19:29:56","slug":"probability-using-venn-diagrams-learn-it-1-update","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probability-using-venn-diagrams-learn-it-1-update\/","title":{"raw":"Probability with Venn Diagrams: Learn It 2","rendered":"Probability with Venn Diagrams: Learn It 2"},"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<h3><strong>Venn Diagrams<\/strong><\/h3>\r\n<p>To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18th century. These illustrations now called\u00a0<strong>Venn Diagrams<\/strong>.<\/p>\r\n<p>A Venn diagram is a visual representation that illustrates the outcomes of a chance experiment.\u00a0<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>Venn diagrams<\/h3>\r\n<p>A <strong>Venn diagram<\/strong> represents each set by a circle\/oval, usually drawn inside of a containing box representing the sample space. Overlapping areas indicate elements common to both sets.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>Basic Venn diagrams can illustrate the interaction of two or three sets.<\/p>\r\n<\/section>\r\n<p>\u00a0\u00a0<img class=\"aligncenter wp-image-5970 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183441\/LI18.2d-300x210.png\" alt=\"A box with an S in the corner containing a blue circle with the letter A inside the circle.\" width=\"300\" height=\"210\" \/><\/p>\r\n<p>To track the outcomes for a specific event, it is useful to indicate the number of outcomes (or proportions) that correspond to that event inside the Venn diagram.<\/p>\r\n<section class=\"textbox example\">Consider the chance experiment where we roll a six-sided die. Let [latex]A = [\/latex] \"the resulting roll is at least 5.\"<br \/>\r\n<strong><br \/>\r\n(a)<\/strong> How many possible outcomes are there in the sample space of [latex]S[\/latex]?<br \/>\r\n[reveal-answer q=\"676340\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"676340\"][latex]6[\/latex][\/hidden-answer]<br \/>\r\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">(b)<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> How many outcomes correspond to the event of [latex]A[\/latex]?<br \/>\r\n[reveal-answer q=\"280861\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"280861\"]2[\/hidden-answer]<\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"><br \/>\r\n<\/span><strong>(c)<\/strong> Draw the Venn diagram with the number of outcomes for each event in the sample space.<br \/>\r\n[reveal-answer q=\"614980\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"614980\"]<img class=\"alignnone size-medium wp-image-5967\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183418\/LI18.2a-300x210.png\" alt=\"The box, labeled S, contains a number 4 written in red. The blue circle labeled A inside the box contains a red number 2.\" width=\"300\" height=\"210\" \/>[\/hidden-answer]<\/section>\r\n<p>There are cases where we might have to consider chance experiments involving multiple events. In such situations, we need to use multiple circles to represent each event. In some special cases, events can overlap, and in these instances, we represent them as intersecting circles, with the counts inside each circle adjusted to account for the overlap.<\/p>\r\n<section class=\"textbox example\">Suppose that a random number is chosen between 1 to 10. Consider the chance experiment where a number is chosen at random.<br \/>\r\n<br \/>\r\nConsider[latex]S=\\{1,2,3,4,5,6,7,8,9,10\\}[\/latex]<br \/>\r\nLet [latex]A = [\/latex] \"the number is even\" and [latex]B = [\/latex] \"the number is at least 8\".<br \/>\r\n<strong><br \/>\r\n(a)<\/strong> How many possible outcomes are there in the sample space of [latex]S[\/latex]?<br \/>\r\n[reveal-answer q=\"359172\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"359172\"][latex]10[\/latex][\/hidden-answer]<br \/>\r\n<strong><br \/>\r\n(b)<\/strong> How many outcomes correspond to the event of [latex]A[\/latex]?<br \/>\r\n[reveal-answer q=\"708219\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"708219\"][latex]5[\/latex][\/hidden-answer]<br \/>\r\n<strong><br \/>\r\n(c) <\/strong>How many outcomes correspond to the event of [latex]B[\/latex]?<br \/>\r\n[reveal-answer q=\"87804\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"87804\"][latex]3[\/latex][\/hidden-answer]<br \/>\r\n<p><strong>(d)<\/strong> Draw the Venn diagram with the number of outcomes for each event in the sample space.<br \/>\r\n[reveal-answer q=\"27821555\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"27821555\"] <img class=\"alignnone size-medium wp-image-5968\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183438\/LI18.2b-300x207.png\" alt=\"A box labeled S contains a red number 4. There is a Venn diagram in the box, with one side labeled A in blue and the other side labeled B in purple. The center of the Venn diagram is green. Inside circle A there is a red number 3, inside the green center there is a red number 2, and inside circle B there is a red number 1.\" width=\"300\" height=\"207\" \/>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]13084[\/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<h3><strong>Venn Diagrams<\/strong><\/h3>\n<p>To visualize the interaction of sets, John Venn in 1880 thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the 18th century. These illustrations now called\u00a0<strong>Venn Diagrams<\/strong>.<\/p>\n<p>A Venn diagram is a visual representation that illustrates the outcomes of a chance experiment.\u00a0<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>Venn diagrams<\/h3>\n<p>A <strong>Venn diagram<\/strong> represents each set by a circle\/oval, usually drawn inside of a containing box representing the sample space. Overlapping areas indicate elements common to both sets.<\/p>\n<p>&nbsp;<\/p>\n<p>Basic Venn diagrams can illustrate the interaction of two or three sets.<\/p>\n<\/section>\n<p>\u00a0\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5970 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183441\/LI18.2d-300x210.png\" alt=\"A box with an S in the corner containing a blue circle with the letter A inside the circle.\" width=\"300\" height=\"210\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183441\/LI18.2d-300x210.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183441\/LI18.2d-65x46.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183441\/LI18.2d-225x158.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183441\/LI18.2d-350x245.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183441\/LI18.2d.png 500w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>To track the outcomes for a specific event, it is useful to indicate the number of outcomes (or proportions) that correspond to that event inside the Venn diagram.<\/p>\n<section class=\"textbox example\">Consider the chance experiment where we roll a six-sided die. Let [latex]A =[\/latex] &#8220;the resulting roll is at least 5.&#8221;<br \/>\n<strong><br \/>\n(a)<\/strong> How many possible outcomes are there in the sample space of [latex]S[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q676340\">Show Answer<\/button><\/p>\n<div id=\"q676340\" class=\"hidden-answer\" style=\"display: none\">[latex]6[\/latex]<\/div>\n<\/div>\n<p>\n<strong style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">(b)<\/strong><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"> How many outcomes correspond to the event of [latex]A[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q280861\">Show Answer<\/button><\/p>\n<div id=\"q280861\" class=\"hidden-answer\" style=\"display: none\">2<\/div>\n<\/div>\n<p><\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"><br \/>\n<\/span><strong>(c)<\/strong> Draw the Venn diagram with the number of outcomes for each event in the sample space.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q614980\">Show Answer<\/button><\/p>\n<div id=\"q614980\" class=\"hidden-answer\" style=\"display: none\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5967\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183418\/LI18.2a-300x210.png\" alt=\"The box, labeled S, contains a number 4 written in red. The blue circle labeled A inside the box contains a red number 2.\" width=\"300\" height=\"210\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183418\/LI18.2a-300x210.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183418\/LI18.2a-65x46.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183418\/LI18.2a-225x158.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183418\/LI18.2a-350x245.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183418\/LI18.2a.png 500w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/div>\n<\/div>\n<\/section>\n<p>There are cases where we might have to consider chance experiments involving multiple events. In such situations, we need to use multiple circles to represent each event. In some special cases, events can overlap, and in these instances, we represent them as intersecting circles, with the counts inside each circle adjusted to account for the overlap.<\/p>\n<section class=\"textbox example\">Suppose that a random number is chosen between 1 to 10. Consider the chance experiment where a number is chosen at random.<\/p>\n<p>Consider[latex]S=\\{1,2,3,4,5,6,7,8,9,10\\}[\/latex]<br \/>\nLet [latex]A =[\/latex] &#8220;the number is even&#8221; and [latex]B =[\/latex] &#8220;the number is at least 8&#8221;.<br \/>\n<strong><br \/>\n(a)<\/strong> How many possible outcomes are there in the sample space of [latex]S[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q359172\">Show Answer<\/button><\/p>\n<div id=\"q359172\" class=\"hidden-answer\" style=\"display: none\">[latex]10[\/latex]<\/div>\n<\/div>\n<p>\n<strong><br \/>\n(b)<\/strong> How many outcomes correspond to the event of [latex]A[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q708219\">Show Answer<\/button><\/p>\n<div id=\"q708219\" class=\"hidden-answer\" style=\"display: none\">[latex]5[\/latex]<\/div>\n<\/div>\n<p>\n<strong><br \/>\n(c) <\/strong>How many outcomes correspond to the event of [latex]B[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q87804\">Show Answer<\/button><\/p>\n<div id=\"q87804\" class=\"hidden-answer\" style=\"display: none\">[latex]3[\/latex]<\/div>\n<\/div>\n<div class=\"wp-nocaption \"><\/div>\n<p><strong>(d)<\/strong> Draw the Venn diagram with the number of outcomes for each event in the sample space.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q27821555\">Show Answer<\/button><\/p>\n<div id=\"q27821555\" class=\"hidden-answer\" style=\"display: none\"> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-5968\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183438\/LI18.2b-300x207.png\" alt=\"A box labeled S contains a red number 4. There is a Venn diagram in the box, with one side labeled A in blue and the other side labeled B in purple. The center of the Venn diagram is green. Inside circle A there is a red number 3, inside the green center there is a red number 2, and inside circle B there is a red number 1.\" width=\"300\" height=\"207\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183438\/LI18.2b-300x207.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183438\/LI18.2b-65x45.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183438\/LI18.2b-225x155.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183438\/LI18.2b-350x241.png 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/09\/26183438\/LI18.2b.png 581w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm13084\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13084&theme=lumen&iframe_resize_id=ohm13084&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":51,"menu_order":5,"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\/3234"}],"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":37,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3234\/revisions"}],"predecessor-version":[{"id":6948,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3234\/revisions\/6948"}],"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\/3234\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=3234"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=3234"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=3234"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=3234"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}