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## Theorem: Bayes' Theorem

Given mutually exclusive and collectively exhaustive events $$A_1,A_2,\ldots,A_n$$ with the probabilities $$p(A_i) > 0$$ for $$i=1,2,\ldots,n$$, and any event $$B$$ with the probability $$p(B) > 0$$, the probabilities of the events $$A_i$$ given $$B$$ can be calculated by the formula

$p(A_i|B)=\frac{p(B|A_i)p(A_i)}{p(B)}=\frac{p(B|A_i)p(A_i)}{\sum_{i=1}^np(B|A_i)p(A_i)},\quad\quad i=1,2,\ldots,n.$

| | | | | created: 2014-03-02 11:40:21 | modified: 2016-03-26 18:38:18 | by: bookofproofs | references: [856]

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