conditional probability
ConditionalProbability
2002-02-18 02:56:03
2005-12-05 08:18:49
Definition
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Let $(\Omega, \borel, \mu)$ be a probability space, and let $X,Y\in\borel$ be events.
The \emph{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.