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| = Conditional Probability = A '''conditional probability''' is the likelihood of an event happening given that another event happens. <<TableOfContents>> ---- == 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''. ---- CategoryRicottone |
