= Conditional Probability =

A '''conditional probability''' is the likelihood of an event happening given that another event happens.

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== Description ==

This is the probability of an event given that some event(s) has (have) already occurred. This is generally notated as ''P(A|B)'' or ''P(A;B)'', where ''B'' has already occurred.

It is generically decomposed as ''P(A|B) = P(A∩B) / P(B)''. Importantly though, '''Bayes theorem''' provides the following decomposition based on [[Statistics/JointProbability|joint probabilities]]:

{{attachment:bayes.svg}}



=== Independence ===

If two events are [[Statistics/JointProbability#Independence|independent]] (notated as ''A⫫B''), 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.



=== 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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