Probability is the chance of occurring an event in a random experiment.<\/p>\n
In an experiment, the probability of happening an event E<\/em><\/strong> is denoted by P(E)<\/strong> and is defined as the ratio of number of elements in the event E<\/strong> to the number of elements in the sample space S.<\/p>\n So, P(E) = n(E)\/n(S),<\/strong> where S<\/em><\/strong> is the sample space.<\/p>\n Example:<\/strong><\/p>\n Find the probability of getting 2<\/strong> heads when 3<\/strong> coins are tossed.<\/p>\n Solution: sample space, S={HHH,HHT,HTH, TTH, THT,HTT, THH, TTT}, n(S) = 8.<\/strong><\/p>\n Let E<\/strong> be the event of getting 2<\/strong> heads, then<\/p>\n E={HHT, HTH, THH}, n(E)=3<\/strong><\/p>\n So,\u00a0 P(E) = n(E)\/n(S) = 3\/8.<\/strong><\/p>\n If E1<\/sub>, E2<\/sub><\/strong> are the events of a sample space S<\/strong> and if E2<\/sub> <\/strong>occurs after the occurrence of E<\/strong>1<\/sub>, then the event of occurrence of E2<\/sub> <\/strong>after the event E1<\/sub> <\/strong>is called conditional event which is denoted by\u00a0 E<\/strong>2<\/sub>|E1<\/sub> or\u00a0 E2\/E1 .<\/strong><\/p>\n Example\u00a0 1:<\/strong><\/span><\/p>\n When three coins are tossed, the event of getting 3<\/strong> heads given that there is at least one head is a conditional event. Example 2:<\/strong><\/span><\/p>\n When two unbiased dice are thrown, getting the sum of 8<\/strong> on the dice given that one of the die showing 2<\/strong> is a conditional event.<\/p>\n If E1<\/sub> <\/strong>and E2<\/sub> <\/strong>are the events of a sample space S, P(E1<\/sub>)\u22600<\/strong> then the probability of E<\/strong>2<\/sub>, after the occurrence of event E<\/strong>1<\/sub>, is called the conditional probability of the event E2<\/sub><\/strong> \u00a0given E1<\/sub> <\/strong>and is denoted by P(E2<\/sub>|E1<\/sub>) \u00a0<\/strong>and we define<\/p>\n P(E2<\/sub>\/E1<\/sub>) = P(E1<\/sub>\u2229E2<\/sub>)\/P(E1<\/sub>).<\/strong><\/p>\n Note:\u00a0 <\/strong>Since the sample space is same for both the events, the above formula can be re-written as,<\/p>\n P(E2<\/sub>\/E1<\/sub>) = n(E1<\/sub>\u2229E2<\/sub>)\/n(E1<\/sub>).<\/strong><\/p>\n Example:<\/strong><\/p>\n Find the probability of getting 2 tails given that there is at least one tail, when 2<\/strong> unbiased coins are tossed.<\/p>\n Solution:<\/strong><\/em><\/p>\n Sample space, S={HH,HT,TH,TT}<\/strong><\/p>\n Let E1<\/sub> <\/strong>be the event of getting at least one tail, then E1<\/sub> = { HT,TH,TT}, n(E1<\/sub>) = 3.;<\/strong><\/p>\n Let E2<\/sub> <\/strong>be the event of getting 2<\/strong> tails, then E1<\/sub>\u2229E2<\/sub> = {TT}, n(E1\u2229E2) =1.<\/strong><\/p>\n So,\u00a0 P(E2<\/sub>\/E1<\/sub>) = n(E1<\/sub>\u2229E2<\/sub>)\/n(E1<\/sub>) = 1\/3<\/strong><\/p>\n 1)\u00a0\u00a0\u00a0\u00a0\u00a0 If E<\/strong> is the event of a sample space S,<\/strong><\/p>\n then P(S|E) = P(E|E) = 1<\/strong><\/p>\n 2)\u00a0\u00a0\u00a0\u00a0\u00a0 If E<\/strong> and F<\/strong> are any two events of a sample space S<\/strong>, such that P(F)\u22600<\/strong>, then<\/p>\n P(E\u2019|F) = 1-P(E|F)<\/strong> SchoolTutoring Academy<\/span> is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Paradise visit: Tutoring in Paradise . <\/a><\/p>\n","protected":false},"excerpt":{"rendered":" Probability: Probability is the chance of occurring an event in a random experiment. In an experiment, the probability of happening an event E is denoted by P(E) and is defined as the ratio of number of elements in the event E to the number of elements in the sample space S. So, P(E) = n(E)\/n(S), […]<\/p>\n","protected":false},"author":19,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[15],"tags":[787,856,1413,2060,2256],"class_list":["post-3200","post","type-post","status-publish","format-standard","hentry","category-probability","tag-how-are-sample-spaces-used","tag-how-does-conditional-probability-work","tag-properties-of-conditional-probability","tag-what-are-properties-of-probability","tag-what-is-conditional-probability"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/3200","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/19"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=3200"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/3200\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=3200"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=3200"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=3200"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Conditional event:<\/h4>\n
\n<\/strong><\/p>\nConditional probability:<\/h4>\n
Properties of conditional probability:<\/h5>\n
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