Suppose an event has two possible outcomes A and B. Then
\[P(A | B)=\frac{P(B | A)P(A)}{P(B)}\]
where \[P(B)=P(B | A)P(A)+P(B | A)P(A')\]
.Proof
\[\begin{equation} \begin{aligned} P(A | B) &= \frac{P(A \cap B)}{P(B)}\\ &= \frac{(B | A)P(A)}{P(B)} \\ &= \frac{(B | A)P(A)}{P(B | A)+P(B | A')} \\ &= \frac{(B | A)P(A)}{P(B | A)P(A)+P(B | A)P(A')} \end{aligned} \end{equation}\]