Statement

Bayes Rule

For two events $A$ and $B$ we have following equality on their conditional probabilities

$$\mathbb{P}[A \vert B] = \frac{\mathbb{P}[B \vert A] \ \mathbb{P}[A]}{\mathbb{P}[B]}$$

Proof

This follows from the definition of conditional probability

$$ \begin{align*} \mathbb{P}[A \vert B] = & \frac{\mathbb{P}[A \cap B]}{\mathbb{P}[B]} & \mbox{from the definition of } \mathbb{P}[A \vert B]\\ = & \frac{\mathbb{P}[B \vert A] \ \mathbb{P}[A]}{\mathbb{P}[B]} & \mbox{as } \mathbb{P}[B \vert A] = \frac{\mathbb{P}[A \cap B]}{\mathbb{P}[A]}. \end{align*} $$