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Conditional Probability

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

The conditional probability of given is defined as

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

(1)

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

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

$\displaystyle \mu(X\vert Y)\mu(Y) = \mu(X\cap Y) = \mu(Y\vert X)\mu(X),$

(2)

and so also

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

(3)