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3.1 Example_Class_2_Solution

# 3.1 Example_Class_2_Solution - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 2 Review Definition of conditional probability For any two events A and B , the conditional probability of A given the occurrence of B is written as P ( A | B ) and is defined as P ( A | B ) = P ( A ∩ B ) P ( B ) provided that P ( B ) > 0. Multiplication theorem (a) For any two events A and B with P ( B ) > 0, P ( A ∩ B ) = P ( B ) P ( A | B ) . (b) For any three events A,B,C with P ( B ∩ C ) > 0, P ( A ∩ B ∩ C ) = P ( C ) P ( B | C ) P ( A | B ∩ C ) . Independence (a) Two events A and B are called independent if and only if P ( A ∩ B ) = P ( A ) P ( B ) . If P ( A ) > 0,then A and B are independent iff P ( B | A ) = P ( B ). (b) The events A 1 ,A 2 , · · · ,A k are (mutually) independent if and only if the probability of the intersection of any combination of them is equal to the product of the probabilities of the corresponding single events. For example, A 1 ,A 2 ,A 3 are independent if and only if P ( A 1 ∩ A 2 ) = P ( A 1 ) P ( A 2 ) P ( A 1 ∩ A 3 ) = P ( A 1 ) P ( A 3 ) P ( A 2 ∩ A 3 ) = P ( A 2 ) P ( A 3 ) P ( A 1 ∩ A 2 ∩ A 3 ) = P ( A 1 ) P ( A 2 ) P ( A 3 ) 1 Law of total probability (a) If 0 < P ( B ) < 1, then P ( A ) = P ( A | B ) P ( B ) + P ( A | B c ) P ( B c ) for any A . (b) If B 1 ,B 2 ,...,B k are mutually exclusive and exhaustive events (i.e. a partition of the sample space), then for any event A , P ( A ) = k X j =1 P ( A | B j ) P ( B j ) where k can also be ∞ . Bayes’ Theorem (Bayes’ rule, Bayes’ law) For any two event A and B with P ( A ) > 0 and P ( B ) > 0, P ( B | A ) = P ( A | B ) P ( B ) P ( A ) . Bayes’ Theorem If B 1 ,B 2 ,...,B k are mutually exclusive and exhaustive events (i.e. a partition of the sample space), and A is any event with P ( A ) > 0, then for any B j , P ( B j | A ) = P ( A | B j ) P ( B j ) P ( A ) = P ( B j ) P ( A | B j ) k X i =1 P ( B i ) P ( A | B i ) . where k can also be ∞ . Problems Problem 1 A and B are two events. Suppose that P ( A | B ) = 0 . 6 ,P ( B | A ) = 0 . 3 and P ( A ∪ B ) = 0 . 72....
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3.1 Example_Class_2_Solution - THE UNIVERSITY OF HONG KONG...

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