conditional probability

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