conditional probability

Let (Ω,𝔅,μ) be a probability spaceMathworldPlanetmath, and let X,Y𝔅 be events.

The conditional probabilityMathworldPlanetmath of X given Y is defined as

μ(X|Y)=μ(XY)μ(Y) (1)

provided μ(Y)>0. (If μ(Y)=0, then μ(X|Y) is not defined.)

If μ(X)>0 and μ(Y)>0, then

μ(X|Y)μ(Y)=μ(XY)=μ(Y|X)μ(X), (2)

and so also

μ(X|Y)=μ(Y|X)μ(X)μ(Y), (3)

which is Bayes’ Theorem.

Title conditional probability
Canonical name ConditionalProbability
Date of creation 2013-03-22 12:21:54
Last modified on 2013-03-22 12:21:54
Owner yark (2760)
Last modified by yark (2760)
Numerical id 8
Author yark (2760)
Entry type Definition
Classification msc 60A99
Related topic ConditionalEntropy
Related topic BayesTheorem
Related topic ConditionalExpectation