{"id":1004,"date":"2023-06-22T01:39:04","date_gmt":"2023-06-22T01:39:04","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/conditional-probabilities-dig-deeper\/"},"modified":"2025-05-10T22:41:58","modified_gmt":"2025-05-10T22:41:58","slug":"conditional-probabilities-dig-deeper","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/conditional-probabilities-dig-deeper\/","title":{"raw":"Conditional Probabilities: Fresh Take","rendered":"Conditional Probabilities: Fresh Take"},"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>Conditional Probabilities<\/h2>\r\n<p>A <strong>conditional probability<\/strong> is calculated based on the assumption that one event has already occurred. Conditional probabilities restrict the total. The new total is indicated after the word GIVEN in the question.<\/p>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question height=\"575\"]1095[\/ohm2_question]<\/p>\r\n<\/section>\r\n<p>Sometimes, when the probability problems are complex, it can be helpful to present the situation in a contingency table. Additionally, some people prefer to use formulas in finding probabilities.<\/p>\r\n<p><strong>Formulas:<\/strong><\/p>\r\n<ul>\r\n\t<li><strong>Conditional probability: <\/strong>For any events [latex]A[\/latex] and [latex]B[\/latex] where [latex]P(B)&gt;0[\/latex],<\/li>\r\n<\/ul>\r\n<p style=\"text-align: center;\">[latex] P(A \\text{ GIVEN } B) = \\frac{P(A \\text{ and } B)}{P(B)} [\/latex]<\/p>\r\n<ul>\r\n\t<li><strong>Independent events: <\/strong>Two events are independent if\u00a0[latex] P(A \\text{ GIVEN } B) = P(A)[\/latex].<\/li>\r\n<\/ul>\r\n<p style=\"padding-left: 60px;\">Alternatively, two events are independent if:<\/p>\r\n<p style=\"padding-left: 60px; text-align: center;\">[latex] P(A \\text{ GIVEN } B) = \\frac{P(A \\text{ and } B)}{P(B)} [\/latex]<\/p>\r\n<p style=\"padding-left: 60px; text-align: center;\">[latex] P(A) = \\frac{P(A \\text{ and } B)}{P(B)} [\/latex]<\/p>\r\n<p style=\"padding-left: 60px; text-align: left;\">Multiply both sides with [latex]P(B)[\/latex]:<\/p>\r\n<p style=\"padding-left: 60px; text-align: center;\">[latex] P(A) \\times P(B) = P(A \\text{ and } B)[\/latex]<\/p>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question height=\"1100\"]1090[\/ohm2_question]<\/p>\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>Conditional Probabilities<\/h2>\n<p>A <strong>conditional probability<\/strong> is calculated based on the assumption that one event has already occurred. Conditional probabilities restrict the total. The new total is indicated after the word GIVEN in the question.<\/p>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm1095\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1095&theme=lumen&iframe_resize_id=ohm1095&source=tnh&show_question_numbers\" width=\"100%\" height=\"575\"><\/iframe><br \/>\n<\/section>\n<p>Sometimes, when the probability problems are complex, it can be helpful to present the situation in a contingency table. Additionally, some people prefer to use formulas in finding probabilities.<\/p>\n<p><strong>Formulas:<\/strong><\/p>\n<ul>\n<li><strong>Conditional probability: <\/strong>For any events [latex]A[\/latex] and [latex]B[\/latex] where [latex]P(B)>0[\/latex],<\/li>\n<\/ul>\n<p style=\"text-align: center;\">[latex]P(A \\text{ GIVEN } B) = \\frac{P(A \\text{ and } B)}{P(B)}[\/latex]<\/p>\n<ul>\n<li><strong>Independent events: <\/strong>Two events are independent if\u00a0[latex]P(A \\text{ GIVEN } B) = P(A)[\/latex].<\/li>\n<\/ul>\n<p style=\"padding-left: 60px;\">Alternatively, two events are independent if:<\/p>\n<p style=\"padding-left: 60px; text-align: center;\">[latex]P(A \\text{ GIVEN } B) = \\frac{P(A \\text{ and } B)}{P(B)}[\/latex]<\/p>\n<p style=\"padding-left: 60px; text-align: center;\">[latex]P(A) = \\frac{P(A \\text{ and } B)}{P(B)}[\/latex]<\/p>\n<p style=\"padding-left: 60px; text-align: left;\">Multiply both sides with [latex]P(B)[\/latex]:<\/p>\n<p style=\"padding-left: 60px; text-align: center;\">[latex]P(A) \\times P(B) = P(A \\text{ and } B)[\/latex]<\/p>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm1090\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1090&theme=lumen&iframe_resize_id=ohm1090&source=tnh&show_question_numbers\" width=\"100%\" height=\"1100\"><\/iframe><br \/>\n<\/section>\n","protected":false},"author":8,"menu_order":23,"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\/1004"}],"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":10,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1004\/revisions"}],"predecessor-version":[{"id":6548,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1004\/revisions\/6548"}],"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\/1004\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1004"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1004"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1004"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1004"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}