Bayes' Rule

Bayes' rule (a.k.a. Bayes' theorem or Bayes law) describes the probability of an event, based on prior knowledge of conditions related to the event.
== Statement of Theorem and Proof ==
From the definition of conditional probability:
P(A|B) = \frac{P(A \cap B)}{P(B)}
P(B|A) = \frac{P(A \cap B)}{P(A)}
Solving for the intersection event probability in the one and substituting the solution in the other gives:
P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{P(B|A)P(A)}{P(B)}