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Let $(\Omega, \borel, \mu)$ be a probability space, and let $X,Y\in\borel$ be events.
The conditional probability of $X$ given $Y$ is defined as \begin{equation} \mu(X|Y) = \frac{\mu(X \cap Y)}{\mu(Y)} \end{equation}provided $\mu(Y)>0$ . (If $\mu(Y)=0$ , then $\mu(X|Y)$ is not defined.)
If $\mu(X)>0$ and $\mu(Y)>0$ , then \begin{equation} \mu(X|Y)\mu(Y) = \mu(X\cap Y) = \mu(Y|X)\mu(X), \end{equation}and so also \begin{equation} \mu(X|Y) = \frac{\mu(Y | X)\mu(X)}{\mu(Y)}, \end{equation}which is Bayes' Theorem.
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