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Unformatted text preview: Suggested answers for exam #1, February 1, 2007 1. In the forthcoming Super Bowl, Bob so strongly believes Chicago will win that he is giving 5:3 odds. Sue likes Indianapoliss chances, and she is giving 3:2 odds. Determine how to bet a total of $ 100 to make the maximum guaranteed profit. a. (10) How much should you bet with Bob and with Sue? Answer: Bet $ x with Bob and $(100- x ) with Sue. If Chicago wins, we have to pay Bob, but collect from Sue, so the profit is- x + 3 2 (100- x ) = 150- 5 2 x. If Indianapolis wins, then we have to pay Sue, but collect from Bob, so the profit is 5 3 x- (100- x ) = 8 3 x- 100 . To have the maximum ensured fixed value for winning, set both equal and solve to obtain 150- 5 2 x = 8 3 x- 100 , or 250 = 31 6 x , or x = 1500 31 , which is slightly less than $50. So, bet $ 1500 31 with Bob and $100- 1500 31 with Sue. b. (2) What is your profit? Answer: Use either equation. If we use the first, it is 150- 5 2 1500 31 , or around $25. c. (3) Bob believe he is making a fair bet. Determine the probability Bob assigns to Chicago winning. Show the computation: just an answer will not suffice. Answer: If Bob believes he is making a fair bet, it means that with these odds he would take either side of the bet. Thus the expected value of his winnings is zero. If p C is the probability he assigns to Chicago, then p I = 1- p C is what he assigns to Indianapolis. For a fair bet, if he bets $1 on Chicago, then the expected value is 0 = E ( X ) = p C (1) + p I (- 5 3 ) = p C (1) + (1- p C )(- 5 3 ) = 8 3 p C- 5 3 , or p C = 5 8 . 2.(10) At your racetrack, you want a guaranteed 5% profit on all money wagered. Just before the three-horse race starts, the windows are closed and you set the odds. $ 6000 was bet on A, $ 3000 on B, and $ 1000 on C. Determine the odds you would set for each horse. Answer: Let x H : 1 be the odds assigned to horse H . The total amount bet is 10 , 000; you want the earnings of (0 . 05)(10 , 000) = 500 . So, just compute what you pay and what you keep; the total should be $500. If A wins, I will have to pay 6000 x A , but I collect 3000 + 1000 = 4000. Therefore, the equation is- 6000 x A + 4000 = 500 , or x A = 3500 6000 = 35 60 = 7 12 ....
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