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assn4-solns - Math 361 Problem set 4 Due 1(1.4.26 Person A...

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Math 361, Problem set 4 Due 9/27/10 1. (1.4.26) Person A tosses a coin and then person B rolls a die. This is re- peated independently until a head or one of the numbers 1 , 2 , 3 , 4 appears, at which time the game is stopped. Person A wins with the head, and B wins with one of the numbers 1 , 2 , 3 , 4. Compute the probability A wins the game. Answer: The probability A wins on his i th coin flip is 1 2 ( 1 6 ) i - 1 ; as he must flip a head on the i th flip, and all other turns he must flip a tail while player B rolls a 4 or 5. If A is the event A wins then P ( A ) = i =1 1 2 1 6 i - 1 = 1 / 2 5 / 6 = 3 5 . A clever alternate method is the following. Suppose we consider the turn when either A or B first wins. Then A wins if and only if he flipped a head on that turn. Let A be the event player A rolls a head that turn, and B be the event player B rolls a 4 or 5. We want P ( A | A B ) = P ( A ) P ( A B ) = 1 / 2 1 - 1 / 6 = 3 5 . Note that this trick doesn’t work so well to calculate the probability that B wins. The reason is that B
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