{"id":3318,"date":"2023-10-03T05:10:04","date_gmt":"2023-10-03T05:10:04","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=3318"},"modified":"2025-06-02T17:45:15","modified_gmt":"2025-06-02T17:45:15","slug":"probability-using-venn-diagrams-learn-it-1-update-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probability-using-venn-diagrams-learn-it-1-update-2\/","title":{"raw":"Probability with Venn Diagrams: Learn It 1","rendered":"Probability with Venn Diagrams: Learn It 1"},"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<h2>Set<\/h2>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>set<\/h3>\r\n\r\nA <strong>set <\/strong>is collection of distinct objects, called elements of the set.<\/div>\r\n<div>\u00a0<\/div>\r\n<div>The elements of a set can be either numbers or objects that have a particular characteristic in common. When defining a set, it is common to use a capital letter to define the set accompanied by the list of elements.<\/div>\r\n<div>\u00a0<\/div>\r\n<div>For example, we can define the set [latex] A = \\{...\\} [\/latex] where the elements of the set are listed inside the curly bracket.<\/div>\r\n<\/section>\r\n<p>A set can contain both a finite or infinite number of elements depending on how the set is defined. The set of real numbers, [latex] R [\/latex] is example of a set commonly used in math that has an infinite number of elements. In probability, we will typically focus on sets that have a finite number of elements.<\/p>\r\n<section class=\"textbox example\">Write out the following set of elements for each of the following sets.<br \/>\r\n<strong><br \/>\r\n<\/strong><strong>(a)<\/strong> The set of letters in the word \"MATH\".<br \/>\r\n<br \/>\r\n[reveal-answer q=\"889637\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"889637\"][latex]A = \\{\\text{M, A, T, H}\\}[\/latex][\/hidden-answer]<br \/>\r\n<strong><br \/>\r\n(b) <\/strong>The set of primary colors.<strong><br \/>\r\n<\/strong><br \/>\r\n[reveal-answer q=\"805354\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"805354\"][latex]A = \\{red, blue, yellow\\}[\/latex][\/hidden-answer]<br \/>\r\n<strong><br \/>\r\n(c)<\/strong>The set of counting numbers from 1 to 10.<br \/>\r\n<br \/>\r\n[reveal-answer q=\"976877\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"976877\"][latex]A = \\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\\}[\/latex][\/hidden-answer]<br \/>\r\n<strong><br \/>\r\n(d) <\/strong>The set of odd numbers from 1 to 6.<br \/>\r\n<br \/>\r\n[reveal-answer q=\"989484\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"989484\"][latex]A = \\{1, 3, 5\\}[\/latex][\/hidden-answer]<\/section>\r\n<p>One of the operations we can perform with a set is to determine the number of elements in the collection. For instance, the set [latex] A = \\{\\text{red, blue, green}\\}[\/latex] contains three elements. With these concepts in mind, sets can be valuable when examined in the context of probability.\u00a0<\/p>\r\n<section class=\"textbox recall\">\r\n<p>The <strong>sample space <\/strong>of a chance experiment is the collection of all possible outcomes for the experiment.<\/p>\r\n<\/section>\r\n<p>When analyzing a chance experiment, both the sample space [latex]S[\/latex] and the event [latex]A[\/latex] consist of sets of outcomes. By understanding the number of elements of these sets, we can calculate the probability of event [latex]A[\/latex] by dividing the number of elements of the sets with the number of elements in the sample space.<\/p>\r\n<section class=\"textbox recall\">\r\n<p class=\"student12ptnumberlist\"><strong>Theoretical probability<\/strong> is the probability that an event will happen based on pure mathematics, not by carrying out an experiment.<\/p>\r\n<p><strong>Notation:<\/strong> [latex]P(\\text{event})[\/latex] indicates \"probability of an event\".<\/p>\r\n<\/section>\r\n<section>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>probability<\/h3>\r\n<p class=\"student12ptnumberlist\">When the outcomes of the sample space are <b>equally likely<\/b>, the <b>probability<\/b> of an event is the number of elements in the event divided by the number of elements in the sample space.<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]P(\\text{event}) = \\dfrac{\\text{number of elements in event}}{\\text{number of all possible elements}}[\/latex]<\/p>\r\n<\/section>\r\n<\/section>\r\n<section class=\"textbox example\">Suppose that carnival wheel is numbered 1 through 9. The wheel is spun at random and a prize is awarded when the number lands on an even number. Let [latex]A = [\/latex] \"the number is even\".<br \/>\r\n<strong><br \/>\r\n(a)<\/strong> Write out the sample space [latex]S[\/latex] using set notation.<br \/>\r\n[reveal-answer q=\"119078\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"119078\"][latex]S = \\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}[\/latex][\/hidden-answer]<strong><br \/>\r\n<br \/>\r\n(b)<\/strong> Write out the event [latex]A[\/latex] using set notation.<br \/>\r\n[reveal-answer q=\"443039\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"443039\"][latex]A = \\{2, 4, 6, 8\\}[\/latex][\/hidden-answer]<strong><br \/>\r\n<br \/>\r\n(c) <\/strong>How many elements are in the set [latex]S[\/latex]?<br \/>\r\n[reveal-answer q=\"231979\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"231979\"][latex]9[\/latex][\/hidden-answer]<strong><br \/>\r\n<br \/>\r\n(d)<\/strong> How many elements are in the set [latex]A[\/latex]?<br \/>\r\n[reveal-answer q=\"407485\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"407485\"][latex]4[\/latex][\/hidden-answer]<br \/>\r\n<br \/>\r\n<strong>(e)<\/strong> Using your results, what is the probability of the event [latex]A[\/latex]?<br \/>\r\n[reveal-answer q=\"874933\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"874933\"][latex]P(A) = \\dfrac{4}{9}[\/latex][\/hidden-answer]<\/section>\r\n<section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]13081[\/ohm2_question]<\/p>\r\n<\/section>\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<h2>Set<\/h2>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>set<\/h3>\n<p>A <strong>set <\/strong>is collection of distinct objects, called elements of the set.<\/div>\n<div>\u00a0<\/div>\n<div>The elements of a set can be either numbers or objects that have a particular characteristic in common. When defining a set, it is common to use a capital letter to define the set accompanied by the list of elements.<\/div>\n<div>\u00a0<\/div>\n<div>For example, we can define the set [latex]A = \\{...\\}[\/latex] where the elements of the set are listed inside the curly bracket.<\/div>\n<\/section>\n<p>A set can contain both a finite or infinite number of elements depending on how the set is defined. The set of real numbers, [latex]R[\/latex] is example of a set commonly used in math that has an infinite number of elements. In probability, we will typically focus on sets that have a finite number of elements.<\/p>\n<section class=\"textbox example\">Write out the following set of elements for each of the following sets.<br \/>\n<strong><br \/>\n<\/strong><strong>(a)<\/strong> The set of letters in the word &#8220;MATH&#8221;.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q889637\">Show Answer<\/button><\/p>\n<div id=\"q889637\" class=\"hidden-answer\" style=\"display: none\">[latex]A = \\{\\text{M, A, T, H}\\}[\/latex]<\/div>\n<\/div>\n<p>\n<strong><br \/>\n(b) <\/strong>The set of primary colors.<strong><br \/>\n<\/strong><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q805354\">Show Answer<\/button><\/p>\n<div id=\"q805354\" class=\"hidden-answer\" style=\"display: none\">[latex]A = \\{red, blue, yellow\\}[\/latex]<\/div>\n<\/div>\n<p>\n<strong><br \/>\n(c)<\/strong>The set of counting numbers from 1 to 10.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q976877\">Show Answer<\/button><\/p>\n<div id=\"q976877\" class=\"hidden-answer\" style=\"display: none\">[latex]A = \\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\\}[\/latex]<\/div>\n<\/div>\n<p>\n<strong><br \/>\n(d) <\/strong>The set of odd numbers from 1 to 6.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q989484\">Show Answer<\/button><\/p>\n<div id=\"q989484\" class=\"hidden-answer\" style=\"display: none\">[latex]A = \\{1, 3, 5\\}[\/latex]<\/div>\n<\/div>\n<\/section>\n<p>One of the operations we can perform with a set is to determine the number of elements in the collection. For instance, the set [latex]A = \\{\\text{red, blue, green}\\}[\/latex] contains three elements. With these concepts in mind, sets can be valuable when examined in the context of probability.\u00a0<\/p>\n<section class=\"textbox recall\">\n<p>The <strong>sample space <\/strong>of a chance experiment is the collection of all possible outcomes for the experiment.<\/p>\n<\/section>\n<p>When analyzing a chance experiment, both the sample space [latex]S[\/latex] and the event [latex]A[\/latex] consist of sets of outcomes. By understanding the number of elements of these sets, we can calculate the probability of event [latex]A[\/latex] by dividing the number of elements of the sets with the number of elements in the sample space.<\/p>\n<section class=\"textbox recall\">\n<p class=\"student12ptnumberlist\"><strong>Theoretical probability<\/strong> is the probability that an event will happen based on pure mathematics, not by carrying out an experiment.<\/p>\n<p><strong>Notation:<\/strong> [latex]P(\\text{event})[\/latex] indicates &#8220;probability of an event&#8221;.<\/p>\n<\/section>\n<section>\n<section class=\"textbox keyTakeaway\">\n<h3>probability<\/h3>\n<p class=\"student12ptnumberlist\">When the outcomes of the sample space are <b>equally likely<\/b>, the <b>probability<\/b> of an event is the number of elements in the event divided by the number of elements in the sample space.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]P(\\text{event}) = \\dfrac{\\text{number of elements in event}}{\\text{number of all possible elements}}[\/latex]<\/p>\n<\/section>\n<\/section>\n<section class=\"textbox example\">Suppose that carnival wheel is numbered 1 through 9. The wheel is spun at random and a prize is awarded when the number lands on an even number. Let [latex]A =[\/latex] &#8220;the number is even&#8221;.<br \/>\n<strong><br \/>\n(a)<\/strong> Write out the sample space [latex]S[\/latex] using set notation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q119078\">Show Answer<\/button><\/p>\n<div id=\"q119078\" class=\"hidden-answer\" style=\"display: none\">[latex]S = \\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}[\/latex]<\/div>\n<\/div>\n<p><strong><\/p>\n<p>(b)<\/strong> Write out the event [latex]A[\/latex] using set notation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q443039\">Show Answer<\/button><\/p>\n<div id=\"q443039\" class=\"hidden-answer\" style=\"display: none\">[latex]A = \\{2, 4, 6, 8\\}[\/latex]<\/div>\n<\/div>\n<p><strong><\/p>\n<p>(c) <\/strong>How many elements are in the set [latex]S[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q231979\">Show Answer<\/button><\/p>\n<div id=\"q231979\" class=\"hidden-answer\" style=\"display: none\">[latex]9[\/latex]<\/div>\n<\/div>\n<p><strong><\/p>\n<p>(d)<\/strong> How many elements are in the set [latex]A[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q407485\">Show Answer<\/button><\/p>\n<div id=\"q407485\" class=\"hidden-answer\" style=\"display: none\">[latex]4[\/latex]<\/div>\n<\/div>\n<p><strong>(e)<\/strong> Using your results, what is the probability of the event [latex]A[\/latex]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q874933\">Show Answer<\/button><\/p>\n<div id=\"q874933\" class=\"hidden-answer\" style=\"display: none\">[latex]P(A) = \\dfrac{4}{9}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm13081\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=13081&theme=lumen&iframe_resize_id=ohm13081&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<\/section>\n","protected":false},"author":51,"menu_order":4,"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\/3318"}],"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":40,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3318\/revisions"}],"predecessor-version":[{"id":6947,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3318\/revisions\/6947"}],"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\/3318\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=3318"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=3318"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=3318"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=3318"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}