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461s11t1s - Math 461 Test 1 Spring 2011 Calculators books...

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Math 461 Test 1, Spring 2011 Calculators, books, notes and extra papers are not allowed on this test! Show all work to qualify for full credits 1. (16 points) (a) Suppose that A 1 , A 2 , A 3 , A 4 , A 5 are independent. Given that P ( A 1 ) = P ( A 2 ) = 1 2 , P ( A 3 ) = P ( A 4 ) = 1 3 , P ( A 5 ) = 1 4 . Find P (( A 1 A 2 ) ( A 3 A 4 ) A 5 ). (b) Suppose X is a geometric random variable with parameter p = 2 3 , find E [(1 - X ) 2 ]. Solution (a) P (( A 1 A 2 ) ( A 3 A 4 ) A 5 ) = P ( A 1 A 2 ) P ( A 3 A 4 ) P ( A 5 ) = ( P ( A 1 ) + P ( A 2 ) - P ( A 1 A 2 ))( P ( A 3 ) + P ( A 4 ) - P ( A 3 A 4 )) P ( A 5 ) = ( 1 2 + 1 2 - 1 4 )( 1 3 + 1 3 - 1 9 ) 1 4 = 5 48 . (b) E [(1 - X ) 2 ] = E [1 - 2 X + X 2 ] = 1 - 2 E [ X ] + E [ X 2 ] = 1 - 2 E [ X ] + (Var( X ) + ( E [ X ]) 2 ) = 1 - 2 · 3 2 + ( 3 4 + 9 4 ) = 1 . 2. (13 points) a 9-card hand is randomly selected from an ordinary deck of 52 cards without replacement. Find the probability that the hand contains all 4 cards of at least 1 of the 13 denominations. Solution Let A 1 be the event that the hand contains all four aces, A 2 be the event the hand contains all four 2’s. The events A 3 , . . . , A 13 are defined similarly. Then we are looking for P (
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