{"id":993,"date":"2023-06-22T01:38:55","date_gmt":"2023-06-22T01:38:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probabilities-of-two-events-dig-deeper\/"},"modified":"2025-05-10T22:36:36","modified_gmt":"2025-05-10T22:36:36","slug":"probabilities-of-two-events-dig-deeper","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probabilities-of-two-events-dig-deeper\/","title":{"raw":"Probability of Compound Events: Fresh Take","rendered":"Probability of Compound Events: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Calculate and interpret probabilities of simple and compound events.<\/li>\r\n\t<li>Understand the concept of mutually exclusive events.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox recall\">\r\n<h3>Probability Properties<\/h3>\r\n<ul>\r\n\t<li>For any two events, [latex]A[\/latex] and [latex]B[\/latex]: [latex]P(A \\text{ or } B) = P(A) + P(B) - P(A \\text{ and } B)[\/latex]<\/li>\r\n\t<li>Two events are mutually exclusive if the probability that they both happen at the same time is zero. That is, if events [latex]A[\/latex] and [latex]B[\/latex] are mutually exclusive, then [latex]P(A \\ \\mathrm{and} \\ B) = 0[\/latex].<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 60px;\">Therefore, [latex]P(A \\text{ or } B) = P(A) + P(B) - P(A \\text{ and } B) = P(A) + P(B) - 0 = P(A) + P(B)[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<p>Watch the following video to explain how you can use the probability property above:<\/p>\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/z-1VvourLsA[\/embed]<\/p>\r\n<p>&nbsp;<\/p>\r\n<\/section>\r\n<p>So, when can we just add probabilities?<\/p>\r\n<section class=\"textbox example\">One card is selected from a deck of cards.<strong><br \/>\r\n<br \/>\r\n(a)<\/strong> What is the probability it is a heart AND a spade?\r\n\r\n<p style=\"text-align: left;\">[reveal-answer q=\"146148\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"146148\"]It is not possible for a card to be a heart and spade at the same time.<br \/>\r\nSo,<\/p>\r\n<p style=\"text-align: center;\">[latex]P(\\text{heart } \\text{and} \\text{ spade})=0[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p><strong>(b)<\/strong> What is the probability it is a heart OR a spade?[reveal-answer q=\"552993\"]Show answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"552993\"]A heart and a spade are mutually exclusive, which we have found in part (a).<br \/>\r\nTherefore:<\/p>\r\n<p style=\"text-align: center;\">[latex]P(\\text{heart } \\text{or} \\text{ spade})=P(\\text{heart})+P(\\text{spade})=\\frac{13}{52}+\\frac{13}{52}=\\frac{26}{52}[\/latex][\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\r\n<p>[embed]https:\/\/www.youtube.com\/embed\/zxhDOvS2c3k[\/embed]<\/p>\r\n<\/section>\r\n<p>&nbsp;<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate and interpret probabilities of simple and compound events.<\/li>\n<li>Understand the concept of mutually exclusive events.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\">\n<h3>Probability Properties<\/h3>\n<ul>\n<li>For any two events, [latex]A[\/latex] and [latex]B[\/latex]: [latex]P(A \\text{ or } B) = P(A) + P(B) - P(A \\text{ and } B)[\/latex]<\/li>\n<li>Two events are mutually exclusive if the probability that they both happen at the same time is zero. That is, if events [latex]A[\/latex] and [latex]B[\/latex] are mutually exclusive, then [latex]P(A \\ \\mathrm{and} \\ B) = 0[\/latex].<\/li>\n<\/ul>\n<p style=\"padding-left: 60px;\">Therefore, [latex]P(A \\text{ or } B) = P(A) + P(B) - P(A \\text{ and } B) = P(A) + P(B) - 0 = P(A) + P(B)[\/latex]<\/p>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<p>Watch the following video to explain how you can use the probability property above:<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex: Probability of Events that are NOT Mutually Exclusive Events\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/z-1VvourLsA?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>&nbsp;<\/p>\n<\/section>\n<p>So, when can we just add probabilities?<\/p>\n<section class=\"textbox example\">One card is selected from a deck of cards.<strong><\/p>\n<p>(a)<\/strong> What is the probability it is a heart AND a spade?<\/p>\n<p style=\"text-align: left;\">\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q146148\">Show answer<\/button><\/p>\n<div id=\"q146148\" class=\"hidden-answer\" style=\"display: none\">It is not possible for a card to be a heart and spade at the same time.<br \/>\nSo,<\/p>\n<p style=\"text-align: center;\">[latex]P(\\text{heart } \\text{and} \\text{ spade})=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><strong>(b)<\/strong> What is the probability it is a heart OR a spade?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q552993\">Show answer<\/button><\/p>\n<div id=\"q552993\" class=\"hidden-answer\" style=\"display: none\">A heart and a spade are mutually exclusive, which we have found in part (a).<br \/>\nTherefore:<\/p>\n<p style=\"text-align: center;\">[latex]P(\\text{heart } \\text{or} \\text{ spade})=P(\\text{heart})+P(\\text{spade})=\\frac{13}{52}+\\frac{13}{52}=\\frac{26}{52}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\">\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Probability of Events that are Mutually Exclusive Events\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/zxhDOvS2c3k?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<\/section>\n<p>&nbsp;<\/p>\n","protected":false},"author":8,"menu_order":15,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":974,"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\/993"}],"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\/8"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/993\/revisions"}],"predecessor-version":[{"id":6543,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/993\/revisions\/6543"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/974"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/993\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=993"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=993"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=993"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=993"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}