{"id":4826,"date":"2021-08-29T12:06:59","date_gmt":"2021-08-29T12:06:59","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4826"},"modified":"2021-11-27T23:09:39","modified_gmt":"2021-11-27T17:39:39","slug":"addition-theorem-of-probability","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/addition-theorem-of-probability\/","title":{"rendered":"Addition Theorem of Probability – Statement and Proof"},"content":{"rendered":"

Here you will learn addition theorem of probability for two and three events with statement and proof.<\/p>\n

Let’s begin –<\/p>\n

Addition Theorem of Probability<\/h2>\n

For two events :<\/strong><\/p>\n

If A and B are two events associated with a random experiment, then<\/p>\n

\n

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

Proof : <\/strong>Let S be the sample space associated with the given random experiment. Suppose the random experiments results in n mutually exclusive ways. Then, S contains n elementary events.<\/p>\n

Let \\(m_1\\), \\(m_2\\) and m be the number of elementary events favourable to A, B and  \\(A \\cap B\\) respectively then,<\/p>\n

P(A) = \\(m_1\\over n\\), P(B) = \\(m_2\\over n\\) and P(\\(A \\cup B\\)) = \\(m\\over n\\)<\/p>\n

The number of elementary events favourable to A only is \\(m_1\\) – m. Similarly, the number of events favourable to B only is \\(m_2\\) – m. Since m events are favourable to both A and B. Therefore, the number of elementary events favourable to A or B or both i.e, \\(A \\cup B\\) is<\/p>\n

\\(m_1\\) – m + \\(m_2\\) – m + m = \\(m_1\\)  + \\(m_2\\) – m<\/p>\n

So, P(\\(A \\cup B\\)) = \\(m_1 + m_2 – m\\over n\\) = \\(m_1\\over n\\) + \\(m_2\\over n\\) – \\(m\\over n\\)<\/p>\n

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

Corollary : <\/strong>If A and B are mutually exclusive events, then<\/p>\n

(\\(A \\cup B\\)) = 0<\/p>\n

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

This is the addition theorem for mutually exclusive events.<\/p>\n

For three events :<\/strong><\/p>\n

If A, B, C are three events associated with a random experiment, then<\/p>\n

\n

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

Corollary : <\/strong>If A, B, C are mutually exclusive events, then<\/p>\n

P(\\(A \\cup B\\)) = P(\\(B \\cup C\\)) = P(\\(A \\cup C\\)) =  P(\\(A \\cap B\\cap C\\)) = 0<\/p>\n

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

This is the addition theorem for three mutually exclusive events.<\/p>\n\n\n

\n
Previous – Formula for Binomial Probability Distribution <\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn addition theorem of probability for two and three events with statement and proof. Let’s begin – Addition Theorem of Probability For two events : If A and B are two events associated with a random experiment, then P(\\(A \\cup B\\)) = P(A) + P(B) – P(\\(A \\cap B\\)) Proof : Let S …<\/p>\n

Addition Theorem of Probability – Statement and Proof<\/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":[545,546],"yoast_head":"\nAddition Theorem of Probability - Statement and Proof - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn addition theorem of probability for two and three events with statement and proof.\" \/>\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\/addition-theorem-of-probability\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Addition Theorem of Probability - 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