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Let $A_1,\ldots,A_n$ be a partition of a sample space where
$P(A_i) > 0$ for all $i$.  Let $B$ be an event with $P(B) > 0$.
Then
$$P(A_i \mid B) = {P(B\mid A_i)P(A_i)\over P(B\mid A_1)P(A_1) + \cdots + P(B\mid A_n)P(A_n)}$$
\bye
