{"id":999,"date":"2023-06-22T01:39:00","date_gmt":"2023-06-22T01:39:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/conditional-probabilities-apply-it-3\/"},"modified":"2025-05-11T23:33:41","modified_gmt":"2025-05-11T23:33:41","slug":"conditional-probabilities-apply-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/conditional-probabilities-apply-it-3\/","title":{"raw":"Conditional Probabilities: Learn It 5","rendered":"Conditional Probabilities: Learn It 5"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Describe and find conditional probabilities.<\/li>\r\n\t<li>Understand the concept of independent events.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Independent and Mutually Exclusive Events<\/h2>\r\n<section class=\"textbox recall\">\r\n<ul>\r\n\t<li>For two events to be\u00a0<strong>independent<\/strong>, the outcome of one event does not impact the outcome of a successive event. Tossing a fair coin or rolling a fair die are often considered independent events. Just because you rolled a [latex]1[\/latex] does not change the probability the next roll will be a [latex]1[\/latex].\r\n\r\n<ul>\r\n\t<li>Sampling with replacement is associated with independent events.<\/li>\r\n\t<li>Sampling without replacement is associated with dependent events.<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li>If two events are\u00a0<strong>mutually exclusive<\/strong>, that means they cannot happen at the same time with a single outcome. Two events are mutually exclusive if the probability of both events happening at the same time is zero. For example, consider flipping a coin. It can land heads up or heads down, but it cannot be both heads up and heads down simultaneously. Thus, heads and tails are mutually-exclusive events.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox proTip\">\r\n<p>Mutually exclusive events are NOT independent events.<\/p>\r\n<p>Mutually exclusive events are events that cannot happen together, while independent events are events where the occurrence of one does not affect the other.<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1576[\/ohm2_question]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]869[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox proTip\">\r\n<p>Event [latex]A[\/latex] and event [latex]B[\/latex] are independent and mutually exclusive events if only if [latex]P(A) = 0 [\/latex] and\/or [latex]P(B) = 0[\/latex]. This is because mutually exclusive means [latex]P(A \\text{ and } B)=0[\/latex] and independent events means [latex]P(A \\text{ and } B) = P(A) \\times P(B)[\/latex].<\/p>\r\n<p>Therefore, [latex]0 = P(A) \\times P(B)[\/latex] and this is only true if [latex]P(A) = 0 [\/latex] and\/or [latex]P(B) = 0[\/latex].<\/p>\r\n<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Describe and find conditional probabilities.<\/li>\n<li>Understand the concept of independent events.<\/li>\n<\/ul>\n<\/section>\n<h2>Independent and Mutually Exclusive Events<\/h2>\n<section class=\"textbox recall\">\n<ul>\n<li>For two events to be\u00a0<strong>independent<\/strong>, the outcome of one event does not impact the outcome of a successive event. Tossing a fair coin or rolling a fair die are often considered independent events. Just because you rolled a [latex]1[\/latex] does not change the probability the next roll will be a [latex]1[\/latex].\n<ul>\n<li>Sampling with replacement is associated with independent events.<\/li>\n<li>Sampling without replacement is associated with dependent events.<\/li>\n<\/ul>\n<\/li>\n<li>If two events are\u00a0<strong>mutually exclusive<\/strong>, that means they cannot happen at the same time with a single outcome. Two events are mutually exclusive if the probability of both events happening at the same time is zero. For example, consider flipping a coin. It can land heads up or heads down, but it cannot be both heads up and heads down simultaneously. Thus, heads and tails are mutually-exclusive events.<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox proTip\">\n<p>Mutually exclusive events are NOT independent events.<\/p>\n<p>Mutually exclusive events are events that cannot happen together, while independent events are events where the occurrence of one does not affect the other.<\/p>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1576\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1576&theme=lumen&iframe_resize_id=ohm1576&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm869\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=869&theme=lumen&iframe_resize_id=ohm869&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox proTip\">\n<p>Event [latex]A[\/latex] and event [latex]B[\/latex] are independent and mutually exclusive events if only if [latex]P(A) = 0[\/latex] and\/or [latex]P(B) = 0[\/latex]. This is because mutually exclusive means [latex]P(A \\text{ and } B)=0[\/latex] and independent events means [latex]P(A \\text{ and } B) = P(A) \\times P(B)[\/latex].<\/p>\n<p>Therefore, [latex]0 = P(A) \\times P(B)[\/latex] and this is only true if [latex]P(A) = 0[\/latex] and\/or [latex]P(B) = 0[\/latex].<\/p>\n<\/section>\n<\/section>\n","protected":false},"author":8,"menu_order":20,"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":"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\/999"}],"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":9,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/999\/revisions"}],"predecessor-version":[{"id":6672,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/999\/revisions\/6672"}],"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\/999\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=999"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=999"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=999"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=999"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}