{"id":3910,"date":"2021-08-10T02:29:49","date_gmt":"2021-08-10T02:29:49","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3910"},"modified":"2021-11-27T23:18:06","modified_gmt":"2021-11-27T17:48:06","slug":"conditional-probability","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/conditional-probability\/","title":{"rendered":"Formula for Conditional Probability"},"content":{"rendered":"

Here, you will learn formula for conditional probability and properties of conditional probability with examples.<\/p>\n

Let’s begin –<\/p>\n

Formula for Conditional Probability<\/h2>\n

Let A and B be two events associated with a random experiment. Then, the probability of occurrence of event A under the condition that B has already occured and P(B) \\(\\ne\\) 0, is called the conditional probability and it is denoted by P(A\/B). Thus, we have<\/p>\n

\n

P(A\/B) = Probability of occurrence of A given that B has already occurred<\/p>\n

P(A\/B) = \\({P(A\\cap B)}\\over P(B)\\) = which is called Conditional Probability of A given B.<\/p>\n<\/blockquote>\n

Similarly, P(B\/A) when P(A) \\(\\ne\\) 0 is defined as the probability of occurrence of event B when A has already occurred.<\/p>\n

\n

P(B\/A) = \\({P(A\\cap B)}\\over P(A)\\) = which is called Conditional Probability of B given A.<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Let there be a bag containing 5 white and 4 red balls. Two balls are drawn from the bag one after the other without replacement.<\/p>\n

Solution<\/span><\/strong> : Consider the following events :<\/p>\n

A = Drawing a white ball in the first draw,<\/p>\n

B = Drawing a red ball in the second draw<\/p>\n

Now,<\/p>\n

P(B\/A) = Probability of drawing a red ball in second draw given that a white ball has already been drawn in the first draw<\/p>\n

\\(\\implies\\) P(B\/A) = Probability of drawing a red ball from a bag containing 4 white and red balls<\/p>\n

\\(\\implies\\) P(B\/A) = \\(4\\over 8\\) = \\(1\\over 2\\)<\/p>\n

For this random experiment P(A\/B) is not meaningful because A cannot occur after the occurence of event B.<\/p>\n

Properties of Conditional Probability<\/strong><\/h4>\n

(i)<\/strong>\u00a0 Let A and B be two events associated with sample space S, then 0 \\(\\le\\) P(A\/B) \\(\\le\\) 1.<\/p>\n

(ii)<\/strong> If A is an event associated with the sample space S of a random experiment, then P(S\/A) = P(A\/A) = 1<\/p>\n

(iii)<\/strong> Let A and B be two events associated with a random experiment and S be the sample space, if C is an evnt such that P(C) \\(\\ne\\) 0, then<\/p>\n

P(\\((A\\cup B)\/C\\)) = P(A\/C) + P(B\/C) – P(\\((A\\cap B)\/C\\))<\/p>\n

In Particular, if A and B are mutually exclusive events, then<\/p>\n

P(\\((A\\cup B)\/C\\)) = P(A\/C) + P(B\/C)<\/p>\n

(iv)\u00a0<\/strong> If A and B are two events associated with a random experiment, the P(A’\/B) = 1 – P(A\/B)<\/p>\n\n\n

\n
Next – Multiplication Theorem Probability<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Probability Basic Concepts <\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn formula for conditional probability and properties of conditional probability with examples. Let’s begin – Formula for Conditional Probability Let A and B be two events associated with a random experiment. Then, the probability of occurrence of event A under the condition that B has already occured and P(B) \\(\\ne\\) 0, is …<\/p>\n

Formula for Conditional Probability<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[14],"tags":[555,556,554],"yoast_head":"\nFormula for Conditional Probability - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn formula for conditional probability and properties of conditional probability with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/conditional-probability\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Formula for Conditional Probability - 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