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.

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