Math170a.test1.solutions170a

# Math170a.test1.solutions170a - T. Liggett Mathematics 170A...

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T. Liggett Mathematics 170A – Midterm 1 Solutions October 18, 2010 (20) 1. Suppose A and B are events with P ( A ) = 3 4 and P ( B ) = 1 3 . (a) What is the largest possible value for P ( A B )? Explain. Solution: P ( A B ) P ( B ) = 1 3 , and this value is attained if B A . (b) What is the smallest possible value for P ( A B )? Explain. Solution: P ( A B ) = P ( A ) + P ( B ) P ( A B ) 3 4 + 1 3 1 = 1 12 , and this value is attained if, for example, Ω = { 1 , 2 , ..., 12 } with each point having weight 1 12 , A = { 1 , ..., 9 } , B = { 9 , ..., 12 } . (10) 2. Three cups, colored red, white and blue respectively, are placed on three saucers, also colored red, white and blue, at random. Find the probability that no cup is on the saucer of the same color. Solution: There are 3! = 6 possible arrangements, and two have no color matches, so the answer is 1 3 .

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## Math170a.test1.solutions170a - T. Liggett Mathematics 170A...

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