{"id":355,"date":"2020-07-08T21:40:23","date_gmt":"2020-07-09T01:40:23","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&#038;p=355"},"modified":"2025-04-25T18:44:20","modified_gmt":"2025-04-25T22:44:20","slug":"3-3-relationship-between-events-and-venn-diagrams","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/3-3-relationship-between-events-and-venn-diagrams\/","title":{"raw":"3.3 Relationship Between Events and Venn Diagrams","rendered":"3.3 Relationship Between Events and Venn Diagrams"},"content":{"raw":"Given events E, A, and B, we can construct new events using the compliment, intersection and union operations:\r\n<ul>\r\n \t<li>The <strong>complement<\/strong> of event E is the event in which E does not occur. The compliment of E is denoted by E<sup>c<\/sup>, or simply \u201cnot E\u201d.<\/li>\r\n \t<li>The <strong>intersection<\/strong> of events A and B is the event in which both A and B occur. The intersection of A and B is denoted by \u00a0A [latex] \\cap[\/latex] B, or A &amp; B, or simply \u201cA and B\u201d.<\/li>\r\n \t<li>The <strong>union<\/strong> of events A and B is the event in which we observe one of the following: A and B both occur, A occurs but B does not, or A does not occur but B does. The union of A and B is denoted by A [latex]\\cup[\/latex] B or simply \u201cA or B\u201d.<\/li>\r\n<\/ul>\r\nTwo events are <strong>mutually exclusive<\/strong> if they do not overlap, i.e., they do not have outcomes in common. If two events A and B are mutually exclusive, they cannot occur at the same time; therefore [latex]A \\cap B=\\varnothing[\/latex] and hence [latex] P(A \\cap B ) = 0[\/latex].\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example: Relationship Between Events and Mutually Exclusive<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSuppose we roll a fair die, so that the sample space is [latex] S[\/latex]= {1, 2, 3, 4, 5, 6}. Consider the following events:\r\n<ul>\r\n \t<li>Observing an even number, A = {2, 4, 6}<\/li>\r\n \t<li>Observing an outcome that is at most 3, B = {1, 2, 3}<\/li>\r\n \t<li>Observing an outcome that is at least 5, C = {5, 6}<\/li>\r\n<\/ul>\r\n<ol>\r\n \t<li>Define and list all possible outcomes of the following events:\r\n<ul>\r\n \t<li>(not A): observing an odd number, not A = {1, 3, 5}<\/li>\r\n \t<li>(A &amp; B): observing an even number that is at most 3, A &amp; B={2}. The only element in the overlap of events A and B, the common element in both the sets of A and B is 2.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Are the events A and B mutually exclusive?<\/li>\r\n<\/ol>\r\n<p style=\"padding-left: 40px;\">No, the overlap is {2}, not an empty set. Therefore, events A and B are NOT mutually exclusive.<\/p>\r\n\u00a0 \u00a0 \u00a0 3. Are the events B and C mutually exclusive?\r\n<p style=\"padding-left: 40px;\">Yes, the don't overlap, i.e., there is no common element in both B and C. Therefore, events B and C are mutually exclusive, i.e., [latex]\\mbox{ B \\&amp; C}=\\varnothing[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\nWe can use a <strong>Venn diagram<\/strong> to show relationships between events. In a Venn diagram, the sample space [latex] S[\/latex] is represented by a rectangle, events are often represented by circles, and events of interest are indicated by a shaded area. The following graphs show the Venn diagrams for the events E, (not E), (A &amp; B), and (A or B) respectively.<a id=\"retfig3.2\"><\/a>\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_2688\" align=\"aligncenter\" width=\"688\"]<img class=\"wp-image-2688 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/venn_diagram.png\" alt=\"Four venn diagrams of events happening in the square space S. 1) The event E, a circle in S. 2) The event not E, outside a circle in S. 3) The event A&amp;B, the overlap between circle A and circle B. 4) The event A or B, all of circle A and circle B. Image description available.\" width=\"688\" height=\"493\" \/> <strong>Figure 3.2<\/strong> Venn Diagrams of Events [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig3.2\">Image Description <\/a><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig3.2\">(See Appendix D Figure 3.2)<\/a>][\/caption]It is straightforward to confirm the basic properties of the probability of an event based on the Venn diagrams:\r\n<ul type=\"disc\">\r\n \t<li>The total area of the rectangle is 1, which means [latex] P(S) = 1[\/latex].<\/li>\r\n \t<li>The probability of the event E is the shaded area, between 0 and 1, which means [latex] 0 \\leq P(E) \\leq 1[\/latex] and [latex] P(\\varnothing) = 0[\/latex].<\/li>\r\n \t<li>If event A is a subset (part of) of event B (See Figure 3.3), denoted by [latex] A \\subseteq B[\/latex], then [latex] P(A) \\leq P(B)[\/latex]. For example, observe in the above diagrams that [latex] A \\: \\&amp; \\: B \\subseteq A [\/latex] and [latex] A \\: \\&amp; \\: B \\subseteq B[\/latex]. From this, it is easy to see that it is always true that [latex] P(A \\: \\&amp; \\: B) \\leq P(A)[\/latex] \u00a0and that [latex] P(A \\: \\&amp; \\: B) \\leq P(B)[\/latex].<a id=\"retfig3.3\"><\/a>\r\n<div>[caption id=\"attachment_2689\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-2689 size-medium\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/AB_subset-300x214.png\" alt=\"A venn diagram showing the square space S with event A as a circle. Inside A is a circle showing event B. Image description available.\" width=\"300\" height=\"214\" \/> <strong>Figure 3.3<\/strong> Event A is a Subset of Event B. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig3.3\">Image Description <\/a><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig3.3\">(See Appendix D Figure 3.3)<\/a>][\/caption]<\/div><\/li>\r\n \t<li>For any two events A and B, since [latex]A\\subseteq (A \\mbox{ or } B)[\/latex], we have [latex]P(A)\\le P(A \\mbox{ or } B)[\/latex]. Similarly, [latex]B\\subseteq (A \\mbox{ or } B)[\/latex], we have [latex]P(B)\\le P(A \\mbox{ or } B)[\/latex].<\/li>\r\n<\/ul>","rendered":"<p>Given events E, A, and B, we can construct new events using the compliment, intersection and union operations:<\/p>\n<ul>\n<li>The <strong>complement<\/strong> of event E is the event in which E does not occur. The compliment of E is denoted by E<sup>c<\/sup>, or simply \u201cnot E\u201d.<\/li>\n<li>The <strong>intersection<\/strong> of events A and B is the event in which both A and B occur. The intersection of A and B is denoted by \u00a0A [latex]\\cap[\/latex] B, or A &amp; B, or simply \u201cA and B\u201d.<\/li>\n<li>The <strong>union<\/strong> of events A and B is the event in which we observe one of the following: A and B both occur, A occurs but B does not, or A does not occur but B does. The union of A and B is denoted by A [latex]\\cup[\/latex] B or simply \u201cA or B\u201d.<\/li>\n<\/ul>\n<p>Two events are <strong>mutually exclusive<\/strong> if they do not overlap, i.e., they do not have outcomes in common. If two events A and B are mutually exclusive, they cannot occur at the same time; therefore [latex]A \\cap B=\\varnothing[\/latex] and hence [latex]P(A \\cap B ) = 0[\/latex].<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example: Relationship Between Events and Mutually Exclusive<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Suppose we roll a fair die, so that the sample space is [latex]S[\/latex]= {1, 2, 3, 4, 5, 6}. Consider the following events:<\/p>\n<ul>\n<li>Observing an even number, A = {2, 4, 6}<\/li>\n<li>Observing an outcome that is at most 3, B = {1, 2, 3}<\/li>\n<li>Observing an outcome that is at least 5, C = {5, 6}<\/li>\n<\/ul>\n<ol>\n<li>Define and list all possible outcomes of the following events:\n<ul>\n<li>(not A): observing an odd number, not A = {1, 3, 5}<\/li>\n<li>(A &amp; B): observing an even number that is at most 3, A &amp; B={2}. The only element in the overlap of events A and B, the common element in both the sets of A and B is 2.<\/li>\n<\/ul>\n<\/li>\n<li>Are the events A and B mutually exclusive?<\/li>\n<\/ol>\n<p style=\"padding-left: 40px;\">No, the overlap is {2}, not an empty set. Therefore, events A and B are NOT mutually exclusive.<\/p>\n<p>\u00a0 \u00a0 \u00a0 3. Are the events B and C mutually exclusive?<\/p>\n<p style=\"padding-left: 40px;\">Yes, the don&#8217;t overlap, i.e., there is no common element in both B and C. Therefore, events B and C are mutually exclusive, i.e., [latex]\\mbox{ B \\& C}=\\varnothing[\/latex].<\/p>\n<\/div>\n<\/div>\n<p>We can use a <strong>Venn diagram<\/strong> to show relationships between events. In a Venn diagram, the sample space [latex]S[\/latex] is represented by a rectangle, events are often represented by circles, and events of interest are indicated by a shaded area. The following graphs show the Venn diagrams for the events E, (not E), (A &amp; B), and (A or B) respectively.<a id=\"retfig3.2\"><\/a><\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_2688\" aria-describedby=\"caption-attachment-2688\" style=\"width: 688px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2688 size-full\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/venn_diagram.png\" alt=\"Four venn diagrams of events happening in the square space S. 1) The event E, a circle in S. 2) The event not E, outside a circle in S. 3) The event A&amp;B, the overlap between circle A and circle B. 4) The event A or B, all of circle A and circle B. Image description available.\" width=\"688\" height=\"493\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/venn_diagram.png 688w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/venn_diagram-300x215.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/venn_diagram-65x47.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/venn_diagram-225x161.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2022\/04\/venn_diagram-350x251.png 350w\" sizes=\"auto, (max-width: 688px) 100vw, 688px\" \/><figcaption id=\"caption-attachment-2688\" class=\"wp-caption-text\"><strong>Figure 3.2<\/strong> Venn Diagrams of Events [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig3.2\">Image Description <\/a><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig3.2\">(See Appendix D Figure 3.2)<\/a>]<\/figcaption><\/figure>\n<p>It is straightforward to confirm the basic properties of the probability of an event based on the Venn diagrams:<\/p>\n<ul type=\"disc\">\n<li>The total area of the rectangle is 1, which means [latex]P(S) = 1[\/latex].<\/li>\n<li>The probability of the event E is the shaded area, between 0 and 1, which means [latex]0 \\leq P(E) \\leq 1[\/latex] and [latex]P(\\varnothing) = 0[\/latex].<\/li>\n<li>If event A is a subset (part of) of event B (See Figure 3.3), denoted by [latex]A \\subseteq B[\/latex], then [latex]P(A) \\leq P(B)[\/latex]. For example, observe in the above diagrams that [latex]A \\: \\& \\: B \\subseteq A[\/latex] and [latex]A \\: \\& \\: B \\subseteq B[\/latex]. From this, it is easy to see that it is always true that [latex]P(A \\: \\& \\: B) \\leq P(A)[\/latex] \u00a0and that [latex]P(A \\: \\& \\: B) \\leq P(B)[\/latex].<a id=\"retfig3.3\"><\/a>\n<div>\n<figure id=\"attachment_2689\" aria-describedby=\"caption-attachment-2689\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2689 size-medium\" src=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/AB_subset-300x214.png\" alt=\"A venn diagram showing the square space S with event A as a circle. Inside A is a circle showing event B. Image description available.\" width=\"300\" height=\"214\" srcset=\"https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/AB_subset-300x214.png 300w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/AB_subset-65x46.png 65w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/AB_subset-225x160.png 225w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/AB_subset-350x249.png 350w, https:\/\/openbooks.macewan.ca\/introstats\/wp-content\/uploads\/sites\/8\/2020\/07\/AB_subset.png 393w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-2689\" class=\"wp-caption-text\"><strong>Figure 3.3<\/strong> Event A is a Subset of Event B. [<a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig3.3\">Image Description <\/a><a href=\"https:\/\/openbooks.macewan.ca\/introstats\/back-matter\/image-description\/#fig3.3\">(See Appendix D Figure 3.3)<\/a>]<\/figcaption><\/figure>\n<\/div>\n<\/li>\n<li>For any two events A and B, since [latex]A\\subseteq (A \\mbox{ or } B)[\/latex], we have [latex]P(A)\\le P(A \\mbox{ or } B)[\/latex]. Similarly, [latex]B\\subseteq (A \\mbox{ or } B)[\/latex], we have [latex]P(B)\\le P(A \\mbox{ or } B)[\/latex].<\/li>\n<\/ul>\n","protected":false},"author":19,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-355","chapter","type-chapter","status-publish","hentry"],"part":327,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/355","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":32,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/355\/revisions"}],"predecessor-version":[{"id":5448,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/355\/revisions\/5448"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/327"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/355\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=355"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=355"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=355"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=355"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}