# Bayes' theorem

## In words...

Bayes' theorem allows the conditional probability of an event $A$ given another event $B$ to be expressed in terms of the conditional probability of $B$ given $A$ and the probabilities of $A$ and $B$.

## In picture...

## In maths...

Given two events $A$ and $B$, we have
$$
P(A|B) = \frac{P(B|A) P (A)}{P(B)}
$$

The proof is a direct consequence of the definition of the

conditional probabilities,
$$
P(A|B) = \frac{P(A\cap B)}{P(B)}
$$
and
$$
P(B|A) = \frac{P(A\cap B)}{P(A)} ,
$$
which implies
$$
P(A|B) P(B) = P(A\cap B) = P(B|A)P(A) .
$$