= Conditional Probability =

A '''conditional probability''' is the likelihood of an event happening given that another event happens. The math notation is ''P(A|B)'', as in the probability of ''A'' given ''B''. 

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== Decomposition ==

{{attachment:decomposition.svg}}



=== Bayes Theorem ===

Bayes combined the decomposition with [[Statistics/JointProbability|joint probability identities]] to arrive at this more solvable theorem.

{{attachment:bayes.svg}}



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== Independence ==

If two events are [[Statistics/Independence|independent]], then probabilities of one do not change from being conditioned on the other.

Put simply, if the conditioning probability is not 0, then:

''P(A|B) = P(A)''

''P(B|A) = P(B)''

A conditioning probability of 0 will cause the conditional probability to be undefined.

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== Conditional Independence ==

If events ''A'' and ''B'' are '''conditionally independent''', then:

''P(A|B,C) = P(A|C)''

''P(A,B|C) = P(A|C) P(B|C)''

This interrelation is sometimes notated as ''((A⫫B)|C)''.



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