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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH2603 Probability Theory Solution to Assignment 3 Problem 26 You have two opponents with whom you alternate play. When- ever you play A , you win with probability p A ; whenever you play B , you win with probability p B , where p B > p A . If your objective is to minimize the number of games you need to win two in a row, should you start with A or with B ? Solution: Let E [ N i ] denote the mean number of games needed if you initially play i ( i = A,B . Suppose we start with A , conditioning on the first two outcomes, we have E [ N A ] = 2 p A p B + p A (1- p B )( E [ N A ] + 2) + (1- p A )( E [ N B ] + 1) Similarly, E [ N B ] = 2 p B p A + p B (1- p A )( E [ N B ] + 2) + (1- p B )( E [ N A ] + 1) Subtracting the second equation from the first equation we have E [ N A ]- E [ N B ] = p A- p B + (1- p A )(1- p B )( E [ N B ]- E [ N A ]) so E [ N A ]- E [ N B ] = p A- p B 1 + (1- p A )(1- p B ) < i.e. E [ N A ] < E [ N B ]. So we should start with A . Problem 69 In the match problem, say that ( i,j ) ,i < j , is a pair if i chooses j ’s hat and j chooses i ’s hat. a. Find the expected number of pairs. b. Let Q n denote the probability that there are no pairs, and derive a recursive formula for Q n in terms of Q j , j < n . c. Use the recursion of part (b) to find Q 8 ....
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- Spring '10
- Probability, Probability theory, Cycle, Qn