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# 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.

Related:

ConditionalEntropy, BayesTheorem, ConditionalExpectation

Type of Math Object:

Definition

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## Mathematics Subject Classification

60A99*no label found*

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