# 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.

Title conditional probability ConditionalProbability 2013-03-22 12:21:54 2013-03-22 12:21:54 yark (2760) yark (2760) 8 yark (2760) Definition msc 60A99 ConditionalEntropy BayesTheorem ConditionalExpectation