Question

Question 9: Let a and z be integers with a > z 2 1, and let p be a real number with 0<p<1. Alexa2 and Zoltan3 play a game consisting of several rounds. In each round, 1, Alexa receives a points with probability p and 0 points with probability 1-p, 2. Zoltan receives z points (with probability 1) We assume that the results of different rounds are independent » Define the event A-"in one round, Alexa receives more points than Zoltan". We say that Alexa is a better player than Zoltan, if Pr(A) >1/2 For which values of p is Alexa a better player than Zoltan? ·Assume that a = 3, z = 2, and p is chosen such that p > 1/2 and p2 < 1/2. (For example, p-(V5-1)/2.) - Is Alexa a better player than Zoltan? - Alexa and Zoltan play a game consisting of two rounds. We consider the total number of points that each player wins during these two rounds. Define the event B-"in two rounds, Alexa receives more points than Zoltan', Prove that Pr(B) < 1/2. (This seems to suggest that Zoltan is a better player than Alexa.) . Let n be a large integer, and assurne that a n + 1, b-n, and p is chosen very close to (but less than) 1. (For example, n- 500 and p- 0.99.) - Is Alexa a better player than Zoltan? - Alexa and Zoltan play a game consisting of n rounds. We consider the total number of points that each player wins during these n rounds. Define the event C- "in n rounds, Alexa receives more points than Zoltan" Prove that Pr(C)- p. (If n -500 and p-0.99, then p0.0066. This seems to suggest that Zoltan is a much better player than Alexa.)

Answer 1

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