\begin{eqnarray} \text{c.f.,} \nonumber \\ & & P(A|B) = \dfrac{P(A \cap B)}{P(B)} \nonumber \\ & & P(B|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) \\ \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) \\ \nonumber \\ \text{by the way } \nonumber \\ P(A|B) & = & \dfrac{P(A \cap B)}{P(B)} \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)} \\ \end{eqnarray}