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HW5-sol (Math 425) - Math 425 Section 008 Homework 5...

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Math 425 – Section 008 Homework 5 – Chapter 4, problems 14, 22(a), 30, 35, 36, 38, 41, 43, 48 Problem 14. Five distinct numbers are randomly distributed to players num- bered 1 to 5. Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, player 1 and 2 compare their num- bers; the winner then compares with player 3, and so on. Let X denote the number of times player 1 is a winner. Find P { X = i } , i = 0 , 1 , . . . , 4 . Solution. Notice that X i if and only if player 1 has the card of highest rank among players 1 , 2 , . . . i + 1. The probability of such an event is 1 i +1 . So P { X i } = 1 i + 1 , i = 0 , 1 , . . . , 4 Now, p ( i ) = P { X = i } = P { X i } - P { X i + 1 } Hence, p (0) = 1 - 1 2 ; p (1) = 1 2 - 1 3 ; p (2) = 1 3 - 1 4 ; p (3) = 1 4 - 1 5 ; p (4) = 1 5 . Problem. 22(a) Suppose that two teams play a series of games that end when one of them has won i games. Suppose that each game played, is, indepedently, won by team A with probability p . Find the expected number of games that are played when i = 2. Show that the number is maximized when p = 1 2 .
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