{"id":992,"date":"2023-06-22T01:38:54","date_gmt":"2023-06-22T01:38:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probabilities-of-two-events-apply-it-2\/"},"modified":"2025-05-11T23:32:00","modified_gmt":"2025-05-11T23:32:00","slug":"probabilities-of-two-events-apply-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probabilities-of-two-events-apply-it-2\/","title":{"raw":"Probability of Compound Events: Apply It 2","rendered":"Probability of Compound Events: Apply It 2"},"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><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Describe the meaning of mutually exclusive and independence using probability&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6721,&quot;3&quot;:{&quot;1&quot;:0},&quot;9&quot;:0,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:2106149},&quot;15&quot;:&quot;Calibri&quot;}\">Describe the meaning of mutually exclusive and independence using probability<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Probability Properties<\/h2>\r\n<section class=\"textbox proTip\">If it is not known whether [latex]A[\/latex] and [latex]B[\/latex] are mutually exclusive, assume they are not until you can show otherwise.<br \/>\r\nYou can always use the following formula to find the probability:\r\n\r\n<p>[latex]P(A \\text{ or } B) = P(A) + P(B) - P(A \\text{ and } B)[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1416[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]17075[\/ohm2_question]<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\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<ul>\r\n\t<li>Two events are independent if [latex]P(A \\text{ and } B )= P(A)\\times P(B)[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>\r\n<\/section>\r\n<p>The availability of a safe and effective vaccine for COVID-19 is a well-recognized additional tool contributing to the control of the COVID-19 pandemic. At the same time, the challenges and efforts needed to rapidly develop, evaluate, and produce this at scale are enormous.[footnote]https:\/\/www.who.int\/emergencies\/diseases\/novel-coronavirus-2019\/global-research-on-novel-coronavirus-2019-ncov\/accelerating-a-safe-and-effective-covid-19-vaccine[\/footnote] Based on all of the research done so far, COVID-19 vaccines can offer added protection to people who had COVID-19, including protection against being hospitalized from a new infection, especially as variants continue to emerge. Now is the time for global solidarity and support, because only together can we overcome the intertwined health, social, and economic impacts of the pandemic. Only then can we protect the health, livelihoods, food security, and nutrition of all people, and ensure that our \u2018new normal\u2019 is a better one.[footnote]https:\/\/www.who.int\/news\/item\/13-10-2020-impact-of-covid-19-on-people's-livelihoods-their-health-and-our-food-systems[\/footnote]<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Calculate and interpret probabilities of simple and compound events.<\/li>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Describe the meaning of mutually exclusive and independence using probability&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:6721,&quot;3&quot;:{&quot;1&quot;:0},&quot;9&quot;:0,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:2106149},&quot;15&quot;:&quot;Calibri&quot;}\">Describe the meaning of mutually exclusive and independence using probability<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Probability Properties<\/h2>\n<section class=\"textbox proTip\">If it is not known whether [latex]A[\/latex] and [latex]B[\/latex] are mutually exclusive, assume they are not until you can show otherwise.<br \/>\nYou can always use the following formula to find the probability:<\/p>\n<p>[latex]P(A \\text{ or } B) = P(A) + P(B) - P(A \\text{ and } B)[\/latex]<br \/>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1416\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1416&theme=lumen&iframe_resize_id=ohm1416&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm17075\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=17075&theme=lumen&iframe_resize_id=ohm17075&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<section class=\"textbox keyTakeaway\">\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<ul>\n<li>Two events are independent if [latex]P(A \\text{ and } B )= P(A)\\times P(B)[\/latex]<\/li>\n<\/ul>\n<\/section>\n<\/section>\n<p>The availability of a safe and effective vaccine for COVID-19 is a well-recognized additional tool contributing to the control of the COVID-19 pandemic. At the same time, the challenges and efforts needed to rapidly develop, evaluate, and produce this at scale are enormous.<a class=\"footnote\" title=\"https:\/\/www.who.int\/emergencies\/diseases\/novel-coronavirus-2019\/global-research-on-novel-coronavirus-2019-ncov\/accelerating-a-safe-and-effective-covid-19-vaccine\" id=\"return-footnote-992-1\" href=\"#footnote-992-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a> Based on all of the research done so far, COVID-19 vaccines can offer added protection to people who had COVID-19, including protection against being hospitalized from a new infection, especially as variants continue to emerge. Now is the time for global solidarity and support, because only together can we overcome the intertwined health, social, and economic impacts of the pandemic. Only then can we protect the health, livelihoods, food security, and nutrition of all people, and ensure that our \u2018new normal\u2019 is a better one.<a class=\"footnote\" title=\"https:\/\/www.who.int\/news\/item\/13-10-2020-impact-of-covid-19-on-people's-livelihoods-their-health-and-our-food-systems\" id=\"return-footnote-992-2\" href=\"#footnote-992-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/p>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-992-1\">https:\/\/www.who.int\/emergencies\/diseases\/novel-coronavirus-2019\/global-research-on-novel-coronavirus-2019-ncov\/accelerating-a-safe-and-effective-covid-19-vaccine <a href=\"#return-footnote-992-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-992-2\">https:\/\/www.who.int\/news\/item\/13-10-2020-impact-of-covid-19-on-people's-livelihoods-their-health-and-our-food-systems <a href=\"#return-footnote-992-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":8,"menu_order":14,"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":"apply_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\/992"}],"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":13,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/992\/revisions"}],"predecessor-version":[{"id":6669,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/992\/revisions\/6669"}],"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\/992\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=992"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=992"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=992"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=992"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}