Bayes' Theorem

\begin{eqnarray*} P(A \mid B) & = & \dfrac{P(A \cap B)}{P(B)} \nonumber \\ P(B \mid A) & = & \dfrac{P(B \cap A)}{P(A)} \nonumber \\ \text{heance } \nonumber \\ P(A \cap B) & = & P(A \mid B) * P(B) \;\; \text{ and } \nonumber \\ P(B \cap A) & = & P(B \mid A) * P(A) \qquad\qquad\qquad\qquad\qquad\qquad\qquad (1) \\ \nonumber \\ \nonumber \\ P(B) & = & P(A \cap B) + P(\neg A \cap B) \nonumber \\ & = & P(B \cap A) + P(B \cap \neg A) \nonumber \\ & = & P(B \mid A) * P(A) + P(B \mid \neg A) * P(\neg A) \qquad\qquad (2) \\ \\ \\ P(A \mid B) & = & \dfrac{P(A \cap B)}{P(B)} \nonumber \\ & = & \dfrac{P(B \cap A)}{P(B)} \;\;\; \text{ from (1) and (2) } \nonumber \\ & = & \dfrac {(1)} {(2)} \nonumber \\ & = & \dfrac {P(B \mid A) * P(A)} {P(B \mid A) * P(A) + P(B \mid \neg A) * P(\neg A)} \qquad\qquad (3) \\ \end{eqnarray*}