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Unformatted text preview: Chapter 3 summary Conditional Probability and Independence Definition and basic formula for conditional probability: Let A,B be two events. If the probability that B occurs is positive, then P ( A | B ) = P ( A B ) P ( B ) . In words, it is the probability that A occurs given that B occurs. We also refer to it as the conditional probability of A given B . In the above formula, we say that we are conditioning on B . Note that it is the event which we condition on whose probability appears in the denominator. Calculation of the probability of an intersection: By cross multiplying the previous formula we get P ( A B ) = P ( A | B ) P ( B ) . This gives us a useful way to calculate the probability that A occurs and B occurs. Of course the inclusion-exclusion formula P ( A B ) = P ( A ) + P ( B )- P ( A B ) also gives a way to calculate the probability of the intersection provided we happen to know P ( A ), P ( B ) and P ( A B ). If we wish to know P ( A B ) we should use whichever of the previous two displayed formulas it is convenient to use. The above formula generalizes to any finite set of events: P ( A 1...
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