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180asolns3 - 180A HW 3 Solutions 1 On a multiple-choice...

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180A HW 3 Solutions October 25, 2006 1) On a multiple-choice quiz with 3 possible answers for each of the 10 questions, what is the expected number of correct answers a student would get just by guessing? Answer Let X denote the number of correct answers guessed. Then note that X Bin (10 , 3) so E [ X ] = 10 / 3 . 2) Two marbles a drawn without replacement from a box that contains 5 red and 5 blue marbles. If they match you win $1 . 10 ; otherwise, you lose $1 . 00 . Calculate the expected value and the standard deviation of the amount you win. Answer Let X denote our winnings, in dollars. We note that P ( X = 1 . 1) = P ( 2 Red Drawn ) + P ( 2 Blue Drawn ) = 2 ( 5 2 )( 5 0 ) ( 10 2 ) = 2 10 45 = 4 9 . Furthermore P ( X = - 1) = 1 - P ( X = 1 . 1) = 5 / 9 (as these are the only two values X can take). Thus, we note that E [ X ] = - 1 P ( X = - 1) + 1 . 1 P ( X = 1 . 1) = - 5 9 + (1 . 1) 4 9 = - 50 90 + 44 90 = - 6 90 We note that E [ X 2 ] = ( - 1) 2 P ( X = - 1) + (1 . 1) 2 P ( X = 1) = 500 900 + 484 900 = 984 900 Finally we note that SD ( X ) = p V ar ( X ) = p E [ X 2 ] - ( E [ X ]) 2 = r 984 900 - 60 900 = r 924 900 3) Jim and Sara play Rock, Paper, Scissors . Assume each round both Jim and Sara choose among rock, paper and scissors with equal probability. (a) What are the chances that the game ends on the sixth round? (b) What are the chances that Sara wins on the fourth round? (c) Suppose Jim and Sara play the game 10 times. Compute the expected number of times that at least three rounds are needed to settle the game.
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