> [!theorem] > > ![[bayes.png|400]] > > Let $A, B$ be [[Event|events]] in a [[Sigma Algebra|sigma algebra]] $\Sigma(S)$ with [[Probability|probabilities]] $P(A), P(B) \ne 0$, then the [[Conditional Probability|conditional probability]]: > $ > P_A(B) = \frac{B(A \cap B)}{P(A)} > = \frac{P_B(A)P(B)}{P(A)} > $ > Let $\mathcal{P} = \{E_1, E_2, \cdots, E_n\}$ be a [[Partition|partition]] of $S$ with each $P(E_i) \ne 0$ and let $A$ and $B$ be as above, then > $ > P_A(B) = \frac{P_B(A)P(B)}{\sum_{i = 1}^{n}P_{E_i}(A)P(E_i)} > $