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The estimate P(A) of the probability of an event A, may change if we are given the fact that some other event B has occurred. The conditional probability of A, given that B has occurred is defined as
Note: The above formula assumes that P(B)>0.
THE MULTIPLICATIVE LAW OF PROBABILITY:Follows directly from the conditional probability definition as
THE LAW OF TOTAL PROBABILITY:Suppose we have a finite (or countably infinite) set of events E1, E2, ... which are (1) pairwise disjoint (mutually exclusive), i.e., Ei ÇEj=f for i¹ j, and (2) exhaustive, i.e., every outcome is contained in one of the events, i.e., Èi Ei = S.
Then for any given event A
P(A) = P(A|E1)P(E1) + P(A|E2)P(E2) + P(A|E3)P(E1) +... = åi P(A|Ei)P(Ei)
BAYES’ THEOREM: First consider a simple version. Suppose A and B are two events having positive probabilities. Then we know that since P(A Ç B) = P(A|B)P(B) = P(B|A)P(A), the conditional probability of A given B is
More generally, suppose E1, E2, E3, ... constitute a set of exhaustive and mutually exclusive events. Then
where the second equation substitutes for P(A) from the law of total probability.
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