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hw1sol - Homework 1 Solution#1 Due 1 Suppose the sample...

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Homework 1 Solution #1 Due 9/12/05 1. Suppose the sample space S = {1,2,3,4,5,6,7,8,9}, A={1,3,5,7}, B={6,7,8,9}, C={2,4,8} and D = {1,5,9}. A. List the elements of the subsets of S that correspond to the following events (1) A B = {6,8,9} (2) ( A B) C = {8} (3) B C = {1,2,3,4,5,8} (4) ( B C) D = {1,5} 5) A C = {2,4,8} (6) ( A C) D = φ B. Suppose it is given that P(A) = 4 9 , P ( B ) = 4 9 , P ( C ) = 3 9 ,P(D) = 3 9 , P ( A B) = 1 9 ,P(A C) = 0, P(A D) = 1 9 ,P(B C) = 1 9 ,P(B D) = 1 9 Intersections of three or four of the subsets A, B, C, D have probability zero.(Why?) Find the probability of each of the subsets in part A. (1) P( A B) = P(B) - P(A B) = 4 9 - 1 9 = 3 9 (2) P[( A B) C] = P(B C) - P(A B C) = 1 9 (3) P( B C) = P( B ) + P(C) - P( B C) = 1 - P(B) - P(C) + P(C) + P(B C) = 1 - P(B) + 2P(C) + P(B C) = 1 - 4 9 + 6 9 + 1 9 = 6 9 (4)
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P[( B C) D] = P[( B D) (C D)] = P( B D) + P(C D) - P( B C D) = P(D) - P(B D) + P(C D) + P(C D) - P(B C D) == P(D) - P(B D) + 2P(C D) - P(B C D) = 3 9 - 1 9 + 0 - 0 = 2 9 (5) P( A C) = P(C) - P(A C) = 3 9 (6) P[( A C) D] = 0 C. (Extra credit) Give a physical problem that will lead to the probabilities in part
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