(related to Theorem: Bayes' Theorem)
By hypothesis, \(p(B) > 0\). It follows from the definition of conditional probability. \[p(A_i|B)=\frac{p(A_i\cap B)}{p(B)}=\frac{p(B\cap A_i)}{p(B)},\quad\quad i=1,2,\ldots,n.\quad\quad( * )\]
Applying the probability of joint events to the nominator of \(( * )\) , we get \[p(A_i|B)=\frac{p(B|A_i)p(A_i)}{p(B)},\quad\quad i=1,2,\ldots,n.\quad\quad( * * )\]
Because by hypothesis all events \(A_1,A_2,\ldots,A_n\) are mutually exclusive and collectively exhaustive with \(p(A_i) > 0\) for \(i=1,2,\ldots,n\), we can apply the law of total probability to the denominator of \( ( * * ) \) and get the required result:
\[p(A_i|B)=\frac{p(B|A_i)p(A_i)}{\sum_{i=1}^np(B|A_i)p(A_i)},\quad\quad i=1,2,\ldots,n.\]
