Two chess players, Vishy and Magnus, play a series of games. Given p, the game results

Chapter 9, Problem 49

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Two chess players, Vishy and Magnus, play a series of games. Given p, the game results are i.i.d. with probability p of Vishy winning, and probability q = 1 p of Magnus winning (assume that each game ends in a win for one of the two players). But p is unknown, so we will treat it as an r.v. To reflect our uncertainty about p, we use the prior p Beta(a, b), where a and b are known positive integers and a 2. (a) Find the expected number of games needed in order for Vishy to win a game (including the win). Simplify fully; your final answer should not use factorials or . (b) Explain in terms of independence vs. conditional independence the direction of the inequality between the answer to (a) and 1 + E(G) for G Geom( a a+b ). (c) Find the conditional distribution of p given that Vishy wins exactly 7 out of the first 10 games.