[[Thereom]] about [[Probability|Probabilities]]: $\huge P(F \mid E) = \frac{P(F)\cdot P(E | F)}{ P(F) \cdot P(E\mid F) + P(\not F)\cdot P(E\mid\not F) } $ ## Generalized Bayes' Thereom Suppose that $E$ is an [[Event]] from a [[Sample Space]] $S$, and that $F_{1}, F_{2}, \dots, F_{n}$ are [[Mutually Exclusivity|Mutually Exclusive]] events such that $\bigcup_{i=1}^n F_{i} = S$. Assume $P(E)\neq 0$ and $\forall i: P(F_{i} )\neq 0$, then: $\huge P(F_{j}|E) = \frac{ P(E|F_{j} \cdot P(F_{j})) }{ \sum_{i=1}^n P(E|F_{i})P(F_{i}) } $