{"id":3730,"date":"2023-10-20T17:17:37","date_gmt":"2023-10-20T17:17:37","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/?post_type=chapter&#038;p=3730"},"modified":"2025-05-17T03:01:54","modified_gmt":"2025-05-17T03:01:54","slug":"probability-using-venn-diagrams-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probability-using-venn-diagrams-fresh-take\/","title":{"raw":"Probability with Venn Diagrams: Fresh Take","rendered":"Probability with Venn Diagrams: Fresh Take"},"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>Venn Diagram<\/h3>\r\n<p>A\u00a0<strong>Venn diagram<\/strong>\u00a0is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events.<\/p>\r\n<section class=\"textbox proTip\">\r\n<p>Let's create Venn diagrams to illustrate <em>A\u00a0<\/em>\u22c3<em>\u00a0B<\/em>,\u00a0<em>A\u00a0<\/em>\u22c2<em>\u00a0B<\/em>, and [latex]A^c[\/latex]\u00a0\u22c2<em>\u00a0B<\/em> .<\/p>\r\n<p><em>A\u00a0<\/em>\u22c3<em>\u00a0B<\/em>\u00a0contains all elements in\u00a0<em>either<\/em> set.<\/p>\r\n<div class=\"qa-wrapper\">\r\n<div id=\"q252649\" class=\"hidden-answer\">\r\n<div class=\"wp-nocaption alignnone size-full wp-image-449\"><img class=\"alignnone size-full wp-image-449\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220329\/ab1.png\" sizes=\"(max-width: 179px) 100vw, 179px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220329\/ab1.png 179w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220329\/ab1-65x45.png 65w\" alt=\"ab1\" width=\"179\" height=\"125\" \/><\/div>\r\n<p><em>A\u00a0<\/em>\u22c2<em>\u00a0B<\/em>\u00a0contains only those elements in both sets \u2013 in the overlap of the circles.<\/p>\r\n<div class=\"wp-nocaption alignnone size-full wp-image-450\"><img class=\"alignnone size-full wp-image-450\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220349\/ab2.png\" sizes=\"(max-width: 180px) 100vw, 180px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220349\/ab2.png 180w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220349\/ab2-65x45.png 65w\" alt=\"ab2\" width=\"180\" height=\"125\" \/><\/div>\r\n<p>[latex]A^c[\/latex] will contain all elements\u00a0<em>not<\/em> in the set A. [latex]A^c[\/latex]\u00a0\u22c2<em>\u00a0B<\/em>\u00a0will contain the elements in set\u00a0<em>B<\/em>\u00a0that are not in set\u00a0<em>A<\/em>.<\/p>\r\n<div class=\"wp-nocaption alignnone size-full wp-image-451\"><img class=\"alignnone size-full wp-image-451\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220403\/ab3.png\" sizes=\"(max-width: 180px) 100vw, 180px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220403\/ab3.png 180w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220403\/ab3-65x45.png 65w\" alt=\"ab3\" width=\"180\" height=\"125\" \/><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Suppose an experiment has the outcomes [latex]1, 2, 3, ..., 12[\/latex]\u00a0where each outcome has an equal chance of occurring.<\/p>\r\n<p>Let event [latex]A = \\{1,2,3,4,5,6\\}[\/latex] and [latex]B = \\{6,7,8,9\\}[\/latex].<\/p>\r\n<p>Then, [latex]A \\text{ and } B = \\{6\\}[\/latex] and [latex]A \\text{ or } B = \\{1,2,3,4,5,6,7,8,9\\}[\/latex]<\/p>\r\n<p>The Venn diagram is as follows:<\/p>\r\n<figure id=\"eip-idm17287840\"><span id=\"id18119489\" data-type=\"media\" data-alt=\"A Venn diagram. An oval representing set A contains the values 1, 2, 3, 4, 5, and 6. An oval representing set B also contains the 6, along with 7, 8, and 9. The values 10, 11, and 12 are present but not contained in either set.\" data-display=\"block\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214409\/fig-ch03_06_01.jpg\" alt=\"A Venn diagram. An oval representing set A contains the values 1, 2, 3, 4, 5, and 6. An oval representing set B also contains the 6, along with 7, 8, and 9. The values 10, 11, and 12 are present but not contained in either set.\" width=\"380\" data-media-type=\"image\/png\" \/><\/span><\/figure>\r\n<\/section>\r\n<p>We can also fill out Venn diagram using the counts the events rather than listing the elements. This will help in calculating the probability afterward.\u00a0<\/p>\r\n<section class=\"textbox example\">\r\n<p>For example, suppose you gave a survey and found that 15 people liked only dogs, 10 people liked only cats, and 40 people liked both dogs and cats. The Venn diagram would consist of two circles, one labeled dogs and one labeled cats.<\/p>\r\n<div class=\"wp-nocaption size-medium wp-image-6439 aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats.jpg\"><img class=\"aligncenter wp-image-6439 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats-300x222.jpg\" sizes=\"(max-width: 300px) 100vw, 300px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats-300x222.jpg 300w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats-65x48.jpg 65w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats-225x166.jpg 225w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats-350x259.jpg 350w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats.jpg 547w\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"300\" height=\"222\" \/><\/a><\/div>\r\n<p>Notice that if you look at the set of those who like dogs, you will see there are 15 who like dogs only, and 40 who like both. The total number of people who like dogs is the sum of 15 and 40, which is 55.<\/p>\r\n<div class=\"wp-nocaption size-medium wp-image-6440 aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4.jpg\"><img class=\"aligncenter wp-image-6440 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4-300x260.jpg\" sizes=\"(max-width: 300px) 100vw, 300px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4-300x260.jpg 300w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4-65x56.jpg 65w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4-225x195.jpg 225w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4-350x304.jpg 350w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4.jpg 607w\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"300\" height=\"260\" \/><\/a><\/div>\r\n<p>Likewise, if you look at the set of those who like cats, you will see there are 10 who like cats only, and 40 who like both. The total number of people who like cats is the sum of 40 and 10, which is 50.<\/p>\r\n<div class=\"wp-nocaption size-medium wp-image-6441 aligncenter\"><img class=\"aligncenter wp-image-6441 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204413\/dogscats5-290x300.jpg\" sizes=\"(max-width: 290px) 100vw, 290px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204413\/dogscats5-290x300.jpg 290w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204413\/dogscats5-65x67.jpg 65w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204413\/dogscats5-225x233.jpg 225w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204413\/dogscats5-350x362.jpg 350w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204413\/dogscats5.jpg 379w\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"290\" height=\"300\" \/><\/div>\r\n<\/section>\r\n<div>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<p><iframe title=\"Solving Problems with Venn Diagrams\" src=\"https:\/\/www.youtube.com\/embed\/MassxXy8iko\" width=\"640\" height=\"360\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\r\n<\/section>\r\n<\/div>\r\n<h3>Calculating Probabilities<\/h3>\r\n<section class=\"textbox example\">\r\n<p>Forty percent of the students at a local college belong to a club, and fifty percent work part-time. Five percent of the students work part-time and belong to a club.<\/p>\r\n<p>Let\u2019s draw a Venn diagram showing the relationships.<\/p>\r\n<p>Let\u00a0<em>C<\/em>\u00a0= student belongs to a club and\u00a0<em>PT<\/em> = student works part-time.<img class=\"aligncenter wp-image-1002 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22013903\/c42cfe1e3e08607ab0b459ae00b3448485a2c022-300x232.jpeg\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"300\" height=\"232\" \/><\/p>\r\n<p>Fill in the necessary parts of the Venn diagram and then use the information to find the following probabilities:<br \/>\r\n<strong>(a)<\/strong>\u00a0[latex]P(C)[\/latex]<br \/>\r\n<strong>(b)<\/strong> [latex]P(PT)[\/latex]<br \/>\r\n<strong>(c)<\/strong> [latex]P(C \\cap PT)[\/latex]<br \/>\r\n<strong>(d)<\/strong> [latex]P(C \\cup PT)[\/latex]<br \/>\r\n<strong>(e)<\/strong> [latex]P(C \\text{ but not } PT)[\/latex]<br \/>\r\n<strong>(f)<\/strong> Are the events [latex]C[\/latex] and [latex]PT[\/latex] mutually exclusive?<\/p>\r\n<p>[reveal-answer q=\"60322\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"60322\"]<\/p>\r\n<p><strong>(a) <\/strong>0.35 + 0.05 = 0.40<strong><br \/>\r\n(b) <\/strong>0.45 + 0.05 = 0.50<br \/>\r\n<strong>(c) <\/strong>0.05<br \/>\r\n<strong>(d) <\/strong>0.35 + 0.05+ 0.45 = 0.85<br \/>\r\n<strong>(e) <\/strong>0.35<br \/>\r\n<strong>(f) <\/strong>No, the intersection of the two events has a probability of 0.05.[\/hidden-answer]<\/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<h3>Venn Diagram<\/h3>\n<p>A\u00a0<strong>Venn diagram<\/strong>\u00a0is a picture that represents the outcomes of an experiment. It generally consists of a box that represents the sample space S together with circles or ovals. The circles or ovals represent events.<\/p>\n<section class=\"textbox proTip\">\n<p>Let&#8217;s create Venn diagrams to illustrate <em>A\u00a0<\/em>\u22c3<em>\u00a0B<\/em>,\u00a0<em>A\u00a0<\/em>\u22c2<em>\u00a0B<\/em>, and [latex]A^c[\/latex]\u00a0\u22c2<em>\u00a0B<\/em> .<\/p>\n<p><em>A\u00a0<\/em>\u22c3<em>\u00a0B<\/em>\u00a0contains all elements in\u00a0<em>either<\/em> set.<\/p>\n<div class=\"qa-wrapper\">\n<div id=\"q252649\" class=\"hidden-answer\">\n<div class=\"wp-nocaption alignnone size-full wp-image-449\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-449\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220329\/ab1.png\" sizes=\"(max-width: 179px) 100vw, 179px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220329\/ab1.png 179w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220329\/ab1-65x45.png 65w\" alt=\"ab1\" width=\"179\" height=\"125\" \/><\/div>\n<p><em>A\u00a0<\/em>\u22c2<em>\u00a0B<\/em>\u00a0contains only those elements in both sets \u2013 in the overlap of the circles.<\/p>\n<div class=\"wp-nocaption alignnone size-full wp-image-450\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-450\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220349\/ab2.png\" sizes=\"(max-width: 180px) 100vw, 180px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220349\/ab2.png 180w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220349\/ab2-65x45.png 65w\" alt=\"ab2\" width=\"180\" height=\"125\" \/><\/div>\n<p>[latex]A^c[\/latex] will contain all elements\u00a0<em>not<\/em> in the set A. [latex]A^c[\/latex]\u00a0\u22c2<em>\u00a0B<\/em>\u00a0will contain the elements in set\u00a0<em>B<\/em>\u00a0that are not in set\u00a0<em>A<\/em>.<\/p>\n<div class=\"wp-nocaption alignnone size-full wp-image-451\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-451\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220403\/ab3.png\" sizes=\"(max-width: 180px) 100vw, 180px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220403\/ab3.png 180w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/276\/2016\/10\/12220403\/ab3-65x45.png 65w\" alt=\"ab3\" width=\"180\" height=\"125\" \/><\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Suppose an experiment has the outcomes [latex]1, 2, 3, ..., 12[\/latex]\u00a0where each outcome has an equal chance of occurring.<\/p>\n<p>Let event [latex]A = \\{1,2,3,4,5,6\\}[\/latex] and [latex]B = \\{6,7,8,9\\}[\/latex].<\/p>\n<p>Then, [latex]A \\text{ and } B = \\{6\\}[\/latex] and [latex]A \\text{ or } B = \\{1,2,3,4,5,6,7,8,9\\}[\/latex]<\/p>\n<p>The Venn diagram is as follows:<\/p>\n<figure id=\"eip-idm17287840\"><span id=\"id18119489\" data-type=\"media\" data-alt=\"A Venn diagram. An oval representing set A contains the values 1, 2, 3, 4, 5, and 6. An oval representing set B also contains the 6, along with 7, 8, and 9. The values 10, 11, and 12 are present but not contained in either set.\" data-display=\"block\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214409\/fig-ch03_06_01.jpg\" alt=\"A Venn diagram. An oval representing set A contains the values 1, 2, 3, 4, 5, and 6. An oval representing set B also contains the 6, along with 7, 8, and 9. The values 10, 11, and 12 are present but not contained in either set.\" width=\"380\" data-media-type=\"image\/png\" \/><\/span><\/figure>\n<\/section>\n<p>We can also fill out Venn diagram using the counts the events rather than listing the elements. This will help in calculating the probability afterward.\u00a0<\/p>\n<section class=\"textbox example\">\n<p>For example, suppose you gave a survey and found that 15 people liked only dogs, 10 people liked only cats, and 40 people liked both dogs and cats. The Venn diagram would consist of two circles, one labeled dogs and one labeled cats.<\/p>\n<div class=\"wp-nocaption size-medium wp-image-6439 aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-6439 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats-300x222.jpg\" sizes=\"(max-width: 300px) 100vw, 300px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats-300x222.jpg 300w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats-65x48.jpg 65w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats-225x166.jpg 225w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats-350x259.jpg 350w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204347\/DogsCats.jpg 547w\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"300\" height=\"222\" \/><\/a><\/div>\n<p>Notice that if you look at the set of those who like dogs, you will see there are 15 who like dogs only, and 40 who like both. The total number of people who like dogs is the sum of 15 and 40, which is 55.<\/p>\n<div class=\"wp-nocaption size-medium wp-image-6440 aligncenter\"><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-6440 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4-300x260.jpg\" sizes=\"(max-width: 300px) 100vw, 300px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4-300x260.jpg 300w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4-65x56.jpg 65w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4-225x195.jpg 225w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4-350x304.jpg 350w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204402\/dogcats4.jpg 607w\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"300\" height=\"260\" \/><\/a><\/div>\n<p>Likewise, if you look at the set of those who like cats, you will see there are 10 who like cats only, and 40 who like both. The total number of people who like cats is the sum of 40 and 10, which is 50.<\/p>\n<div class=\"wp-nocaption size-medium wp-image-6441 aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-6441 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204413\/dogscats5-290x300.jpg\" sizes=\"(max-width: 290px) 100vw, 290px\" srcset=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204413\/dogscats5-290x300.jpg 290w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204413\/dogscats5-65x67.jpg 65w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204413\/dogscats5-225x233.jpg 225w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204413\/dogscats5-350x362.jpg 350w, https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/01\/25204413\/dogscats5.jpg 379w\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"290\" height=\"300\" \/><\/div>\n<\/section>\n<div>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<p><iframe loading=\"lazy\" title=\"Solving Problems with Venn Diagrams\" src=\"https:\/\/www.youtube.com\/embed\/MassxXy8iko\" width=\"640\" height=\"360\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/section>\n<\/div>\n<h3>Calculating Probabilities<\/h3>\n<section class=\"textbox example\">\n<p>Forty percent of the students at a local college belong to a club, and fifty percent work part-time. Five percent of the students work part-time and belong to a club.<\/p>\n<p>Let\u2019s draw a Venn diagram showing the relationships.<\/p>\n<p>Let\u00a0<em>C<\/em>\u00a0= student belongs to a club and\u00a0<em>PT<\/em> = student works part-time.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1002 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22013903\/c42cfe1e3e08607ab0b459ae00b3448485a2c022-300x232.jpeg\" alt=\"Appropriate alternative text can be found in the description above.\" width=\"300\" height=\"232\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22013903\/c42cfe1e3e08607ab0b459ae00b3448485a2c022-300x232.jpeg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22013903\/c42cfe1e3e08607ab0b459ae00b3448485a2c022-65x50.jpeg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22013903\/c42cfe1e3e08607ab0b459ae00b3448485a2c022-225x174.jpeg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22013903\/c42cfe1e3e08607ab0b459ae00b3448485a2c022-350x270.jpeg 350w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22013903\/c42cfe1e3e08607ab0b459ae00b3448485a2c022.jpeg 488w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Fill in the necessary parts of the Venn diagram and then use the information to find the following probabilities:<br \/>\n<strong>(a)<\/strong>\u00a0[latex]P(C)[\/latex]<br \/>\n<strong>(b)<\/strong> [latex]P(PT)[\/latex]<br \/>\n<strong>(c)<\/strong> [latex]P(C \\cap PT)[\/latex]<br \/>\n<strong>(d)<\/strong> [latex]P(C \\cup PT)[\/latex]<br \/>\n<strong>(e)<\/strong> [latex]P(C \\text{ but not } PT)[\/latex]<br \/>\n<strong>(f)<\/strong> Are the events [latex]C[\/latex] and [latex]PT[\/latex] mutually exclusive?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q60322\">Show Answer<\/button><\/p>\n<div id=\"q60322\" class=\"hidden-answer\" style=\"display: none\">\n<p><strong>(a) <\/strong>0.35 + 0.05 = 0.40<strong><br \/>\n(b) <\/strong>0.45 + 0.05 = 0.50<br \/>\n<strong>(c) <\/strong>0.05<br \/>\n<strong>(d) <\/strong>0.35 + 0.05+ 0.45 = 0.85<br \/>\n<strong>(e) <\/strong>0.35<br \/>\n<strong>(f) <\/strong>No, the intersection of the two events has a probability of 0.05.<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":12,"menu_order":10,"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":"fresh_take","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\/3730"}],"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":18,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3730\/revisions"}],"predecessor-version":[{"id":6940,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/3730\/revisions\/6940"}],"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\/3730\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=3730"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=3730"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=3730"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=3730"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}