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W2-slides2 - Lecture 5 Outline and Examples(3.2 and 3.3 in...

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Lecture 5- Outline and Examples ( § 3.2 and § 3.3 in Ross) Conditional Probability continued Multiplication rule Rule of average conditional probabilities partition of S Bayes’ formula
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Conditional Probability (Ross § 3.2) If P ( B ) > 0, then the probability that event A occurs given that B has occurred is P ( A | B ) = P ( A B ) P ( B ) .
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Multiplication rule ( Ross § 3.2) For any two events A and B with P ( B ) > 0, P ( A B ) = P ( B ) P ( A | B ) . Generalized rule: P ( A 1 A 2 ∩· · · A n ) = P ( A 1 ) P ( A 2 | A 1 ) P ( A 3 | A 1 A 2 ) · · · P ( A n | A 1 ∩· · · A n - 1 ) .
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Rule of Average Conditional Probabilities For any events A and B with P ( B ) > 0 write P ( A ) = P ( A B ) + P ( A B c ) addition rule = P ( A | B ) P ( B ) + P ( A | B c ) P ( B c ) multiplication rule
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Rule of Average Conditional Probabilities For any events A and B with P ( B ) > 0 write P ( A ) = P ( A B ) + P ( A 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.
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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 A j = for all i 6 = j .
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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 A j = 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 ) > 0 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 ) .
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Average conditional probabilities-example 1 Example. Sampling without replacing.
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