PlanetMath: conditional probability

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conditional probability

(Definition)

Let $(\Omega, \mathfrak{B}, \mu)$ be a probability space, and let $X,Y\in\mathfrak{B}$ be events.

The conditional probability of given is defined as

$\displaystyle \mu(X\vert Y) = \frac{\mu(X \cap Y)}{\mu(Y)}$

(1)

provided $\mu(Y)>0$ . (If $\mu(Y)=0$ , then $\mu(X\vert Y)$ is not defined.)

If $\mu(X)>0$ and $\mu(Y)>0$ , then

$\displaystyle \mu(X\vert Y)\mu(Y) = \mu(X\cap Y) = \mu(Y\vert X)\mu(X),$

(2)

and so also

$\displaystyle \mu(X\vert Y) = \frac{\mu(Y \vert X)\mu(X)}{\mu(Y)},$

(3)

"conditional probability" is owned by yark. [ full author list (2) | owner history (1) ]

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Attachments:: probability conditioning on a sigma algebra (Definition) by CWoo
regular conditional probability (Definition) by CWoo

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Cross-references: Bayes theorem, events, probability space
There are 9 references to this entry.

This is version 5 of conditional probability, born on 2002-02-18, modified 2005-12-05.
Object id is 2097, canonical name is ConditionalProbability.
Accessed 8773 times total.

Classification:

60A99 (Probability theory and stochastic processes :: Foundations of probability theory :: Miscellaneous)

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