Theorem (Bayes Theorem)

Let $B_1,\cdots,B_n$ be disjoint events such that $\cup_{i=1}^nB_i=\Omega$ and let $A\subset\Omega$ then $$P(B_k|A)=\frac{P(A|B_k)P(B_k)}{\sum_{i=1}^nP(A|B_i)P(B_i)}$$