{"id":6261,"date":"2021-10-11T02:17:51","date_gmt":"2021-10-10T20:47:51","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6261"},"modified":"2022-01-16T17:05:06","modified_gmt":"2022-01-16T11:35:06","slug":"total-probability-theorem","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/total-probability-theorem\/","title":{"rendered":"Total Probability Theorem – Definition and Examples"},"content":{"rendered":"

Here you will learn law of total probability theorem with examples.<\/p>\n

Let’s begin –<\/p>\n

Total Probability Theorem (Law of Total Probability)<\/h2>\n

Let an event A of an experiment occurs with its n mutually exclusive and exhaustive events \\(B_1,B_2,B_3….B_n\\) then total probability of occurrence of event A is<\/p>\n

\n

P(A) = \\(P(AB_1)\\) + \\(P(AB_2)\\) +……..+ \\(P(AB_n)\\)<\/p>\n

P(A) = P(\\(B_1\\))P(\\(A\/B_1\\)) + P(\\(B_2\\))P(\\(A\/B_2\\)) +……..+ P(\\(B_n\\))P(\\(A\/B_n\\))<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : A bag contains 4 red and 3 black balls. A second bag contains 2 red and 4 black balls. One bag is selected at random. From the selected bag, one ball is drawn. Find the probability that the ball drawn is red.<\/p>\n

Solution<\/span><\/strong> : A red ball can be drawn in two mutually exclusive ways.<\/p>\n

(I) Selecting bag I and then drawing a red ball from it.<\/p>\n

(II) Selecting bag II and then drawing a red ball from it.<\/p>\n

Let \\(E_1\\), \\(E_2\\) and A denote the events defined as follows :<\/p>\n

\\(E_1\\) = Selecting bag I,<\/p>\n

\\(E_2\\) = Selecting bag II<\/p>\n

A = Drawing a red ball<\/p>\n

Since one of the two bags is selected randomly.<\/p>\n

\\(\\therefore\\)  P\\(E_1\\) = \\(1\\over 2\\) and P\\(E_2\\) = \\(1\\over 2\\)<\/p>\n

Now, P(A\/\\(E_1\\)) = Probability of drawing a red ball when the first bag has been chosen.<\/p>\n

= \\(4\\over 7\\)                        [ because First bag contains 4 red and 3 black balls ]<\/p>\n

and, P(A\/\\(E_2\\)) = Probability of drawing a red ball when the second bag has been chosen.<\/p>\n

= \\(2\\over 6\\)                        [ because Second bag contains 2 red and 4 black balls ]<\/p>\n

Using the law of total probability, we have<\/p>\n

Required Probabilty = P(A) = P(\\(E_1\\)) P(A\/\\(E_1\\)) + P(\\(E_2\\)) P(A\/\\(E_2\\))<\/p>\n

= \\(1\\over 2\\) \\(\\times\\) \\(4\\over 7\\) + \\(1\\over 2\\) \\(\\times\\) \\(2\\over 6\\) = \\(19\\over 42\\)<\/p>\n

Example<\/strong><\/span> : A purse contains 4 copper and 3 silver coins and another purse contains 6 copper and 2 silver coins. One coin is drawn from any one of the these two purses. The probability that it is a copper coin is-<\/p>\n

Solution<\/span><\/strong> : Let A = event of selecting first purse<\/p>\n

B = event of selecting second purse<\/p>\n

C = event of selecting copper coin<\/p>\n

Then given event has two disjoint cases : AC and BC<\/p>\n

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

= \\(1\\over 2\\).\\(4\\over 7\\) + \\(1\\over 2\\).\\(6\\over 8\\) = \\(37\\over 56\\)<\/p>\n\n\n

\n
Next – Formula for Bayes Theorem \u2013 Definition & Example<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Dependent and Independent Events in Probability<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn law of total probability theorem with examples. Let’s begin – Total Probability Theorem (Law of Total Probability) Let an event A of an experiment occurs with its n mutually exclusive and exhaustive events \\(B_1,B_2,B_3….B_n\\) then total probability of occurrence of event A is P(A) = \\(P(AB_1)\\) + \\(P(AB_2)\\) +……..+ \\(P(AB_n)\\) P(A) …<\/p>\n

Total Probability Theorem – Definition and Examples<\/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":[540,539,541],"yoast_head":"\nTotal Probability Theorem - Definition and Examples<\/title>\n<meta name=\"description\" content=\"In this post you will learn law of total probability theorem 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\/total-probability-theorem\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Total Probability Theorem - Definition and Examples\" \/>\n<meta property=\"og:description\" content=\"In this post you will learn law of total probability theorem with examples.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathemerize.com\/total-probability-theorem\/\" \/>\n<meta property=\"og:site_name\" content=\"Mathemerize\" \/>\n<meta property=\"article:published_time\" content=\"2021-10-10T20:47:51+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-01-16T11:35:06+00:00\" \/>\n<meta name=\"author\" content=\"mathemerize\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"mathemerize\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathemerize.com\/total-probability-theorem\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/total-probability-theorem\/\"},\"author\":{\"name\":\"mathemerize\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\"},\"headline\":\"Total Probability Theorem – Definition and Examples\",\"datePublished\":\"2021-10-10T20:47:51+00:00\",\"dateModified\":\"2022-01-16T11:35:06+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathemerize.com\/total-probability-theorem\/\"},\"wordCount\":416,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"keywords\":[\"law of total probability\",\"total probability theorem\",\"total probability theorem examples\"],\"articleSection\":[\"Probability\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathemerize.com\/total-probability-theorem\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathemerize.com\/total-probability-theorem\/\",\"url\":\"https:\/\/mathemerize.com\/total-probability-theorem\/\",\"name\":\"Total Probability Theorem - Definition and Examples\",\"isPartOf\":{\"@id\":\"https:\/\/mathemerize.com\/#website\"},\"datePublished\":\"2021-10-10T20:47:51+00:00\",\"dateModified\":\"2022-01-16T11:35:06+00:00\",\"description\":\"In this post you will learn law of total probability theorem with examples.\",\"breadcrumb\":{\"@id\":\"https:\/\/mathemerize.com\/total-probability-theorem\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathemerize.com\/total-probability-theorem\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathemerize.com\/total-probability-theorem\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathemerize.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Total Probability Theorem – Definition and Examples\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathemerize.com\/#website\",\"url\":\"https:\/\/mathemerize.com\/\",\"name\":\"Mathemerize\",\"description\":\"Maths Tutorials - Study Math Online\",\"publisher\":{\"@id\":\"https:\/\/mathemerize.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathemerize.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathemerize.com\/#organization\",\"name\":\"Mathemerize\",\"url\":\"https:\/\/mathemerize.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"contentUrl\":\"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1\",\"width\":140,\"height\":96,\"caption\":\"Mathemerize\"},\"image\":{\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/\"},\"sameAs\":[\"https:\/\/www.instagram.com\/mathemerize\/\"]},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df\",\"name\":\"mathemerize\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/mathemerize.com\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g\",\"caption\":\"mathemerize\"},\"sameAs\":[\"https:\/\/mathemerize.com\"],\"url\":\"https:\/\/mathemerize.com\/author\/mathemerize\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Total Probability Theorem - Definition and Examples","description":"In this post you will learn law of total probability theorem with examples.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathemerize.com\/total-probability-theorem\/","og_locale":"en_US","og_type":"article","og_title":"Total Probability Theorem - Definition and Examples","og_description":"In this post you will learn law of total probability theorem with examples.","og_url":"https:\/\/mathemerize.com\/total-probability-theorem\/","og_site_name":"Mathemerize","article_published_time":"2021-10-10T20:47:51+00:00","article_modified_time":"2022-01-16T11:35:06+00:00","author":"mathemerize","twitter_card":"summary_large_image","twitter_misc":{"Written by":"mathemerize","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathemerize.com\/total-probability-theorem\/#article","isPartOf":{"@id":"https:\/\/mathemerize.com\/total-probability-theorem\/"},"author":{"name":"mathemerize","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df"},"headline":"Total Probability Theorem – Definition and Examples","datePublished":"2021-10-10T20:47:51+00:00","dateModified":"2022-01-16T11:35:06+00:00","mainEntityOfPage":{"@id":"https:\/\/mathemerize.com\/total-probability-theorem\/"},"wordCount":416,"commentCount":0,"publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"keywords":["law of total probability","total probability theorem","total probability theorem examples"],"articleSection":["Probability"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathemerize.com\/total-probability-theorem\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathemerize.com\/total-probability-theorem\/","url":"https:\/\/mathemerize.com\/total-probability-theorem\/","name":"Total Probability Theorem - Definition and Examples","isPartOf":{"@id":"https:\/\/mathemerize.com\/#website"},"datePublished":"2021-10-10T20:47:51+00:00","dateModified":"2022-01-16T11:35:06+00:00","description":"In this post you will learn law of total probability theorem with examples.","breadcrumb":{"@id":"https:\/\/mathemerize.com\/total-probability-theorem\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathemerize.com\/total-probability-theorem\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathemerize.com\/total-probability-theorem\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathemerize.com\/"},{"@type":"ListItem","position":2,"name":"Total Probability Theorem – Definition and Examples"}]},{"@type":"WebSite","@id":"https:\/\/mathemerize.com\/#website","url":"https:\/\/mathemerize.com\/","name":"Mathemerize","description":"Maths Tutorials - Study Math Online","publisher":{"@id":"https:\/\/mathemerize.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathemerize.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/mathemerize.com\/#organization","name":"Mathemerize","url":"https:\/\/mathemerize.com\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/","url":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","contentUrl":"https:\/\/i1.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/05\/logo.png?fit=140%2C96&ssl=1","width":140,"height":96,"caption":"Mathemerize"},"image":{"@id":"https:\/\/mathemerize.com\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.instagram.com\/mathemerize\/"]},{"@type":"Person","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/104c8bc54f90618130a6665299bc55df","name":"mathemerize","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/mathemerize.com\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/f0649d8b9c9f4ba7f1682b12d040d2a3?s=96&d=mm&r=g","caption":"mathemerize"},"sameAs":["https:\/\/mathemerize.com"],"url":"https:\/\/mathemerize.com\/author\/mathemerize\/"}]}},"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6261"}],"collection":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/comments?post=6261"}],"version-history":[{"count":10,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6261\/revisions"}],"predecessor-version":[{"id":9564,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/posts\/6261\/revisions\/9564"}],"wp:attachment":[{"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/media?parent=6261"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/categories?post=6261"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathemerize.com\/wp-json\/wp\/v2\/tags?post=6261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}