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Unformatted text preview: Lecture 5 Outline and Examples ( Â§ 3.2 and Â§ 3.3 in Ross) Conditional Probability continued Multiplication rule Rule of average conditional probabilities partition of S Bayesâ€™ formula Conditional Probability (Ross Â§ 3.2) If P ( B ) > 0, then the probability that event A occurs given that B has occurred is P ( A  B ) = P ( A âˆ© B ) P ( B ) . Multiplication rule ( Ross Â§ 3.2) For any two events A and B with P ( B ) > 0, P ( A âˆ© B ) = P ( B ) P ( A  B ) . Generalized rule: P ( A 1 âˆ© A 2 âˆ©Â·Â·Â· A n ) = P ( A 1 ) P ( A 2  A 1 ) P ( A 3  A 1 âˆ© A 2 ) Â·Â·Â· P ( A n  A 1 âˆ©Â·Â·Â· A n 1 ) . Rule of Average Conditional Probabilities For any events A and B with P ( B ) > 0 write P ( A ) = P ( A âˆ© B ) + P ( A âˆ© B c ) addition rule = P ( A  B ) P ( B ) + P ( A  B c ) P ( B c ) multiplication rule Rule of Average Conditional Probabilities For any events A and B with P ( B ) > 0 write P ( A ) = P ( A âˆ© B ) + P ( A âˆ© B c ) addition rule = P ( A  B ) P ( B ) + P ( A  B c ) P ( B c ) multiplication rule We partitioned the sample set S into B S B c and then looked at the intersection of A with each component of the partition. General Rule of Average Conditional Probabilities A collection of events A 1 , A 2 , . . . , A n defined on S is called a partition of S , if n [ i =1 A i = S , and A i âˆ© A j = âˆ… for all i 6 = j . General Rule of Average Conditional Probabilities A collection of events A 1 , A 2 , . . . , A n defined on S is called a partition of S , if n [ i =1 A i = S , and A i âˆ© A j = âˆ… for all i 6 = j . In general, Rule of Average Conditional Probabilities says: Theorem. Let B 1 , B 2 , . . . , B n be a partition of S. Assume that P ( B i ) > for all i. Then for any event A, P ( A ) = P ( A  B 1 ) P ( B 1 ) + . . . + P ( A  B n ) P ( B n ) In words: the overall probability P ( A ) is the weighted average of the conditional probabilities P ( A  B i ) with weights P ( B i ) . Average conditional probabilitiesexample 1...
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 Spring '10
 RohiniKumar
 Conditional Probability, Probability, average conditional probabilities

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