conditional probability

Let $(\Omega,\mathfrak{B},\mu)$ be a probability space, and let $X,Y\in\mathfrak{B}$ be events.

The conditional probability of $X$ given $Y$ is defined as

$\mu(X|Y)=\frac{\mu(X\cap Y)}{\mu(Y)}$

(1)

provided $\mu(Y)>0$ . (If $\mu(Y)=0$ , then $\mu(X|Y)$ is not defined.)

If $\mu(X)>0$ and $\mu(Y)>0$ , then

$\mu(X|Y)\mu(Y)=\mu(X\cap Y)=\mu(Y|X)\mu(X),$

(2)

and so also

$\mu(X|Y)=\frac{\mu(Y|X)\mu(X)}{\mu(Y)},$

(3)

which is Bayes’ Theorem.