Gambles are independent, and each one results in the playerbeing equally likely

Question

Gambles are independent, and each one results in the playerbeing equally likely to win or lose 1 unit. Let W denote the netwinnings of a gambler whose strategy is to stop gamblingimmediately after his first win. Find
(a) P{W > 0} (b) P{W < 0} (c) P E[W] Gambles are independent, and each one results in the playerbeing equally likely to win or lose 1 unit. Let W denote the netwinnings of a gambler whose strategy is to stop gamblingimmediately after his first win. Find
(a) P{W > 0} (b) P{W < 0} (c) P E[W]

Explanation / Answer

(a)   Let X be the number of gamesplayed.        If X=1, then W =1        If X=2, then W = -1 + 1 =0        If X=3, then W = -1 -1 +1= -1        ...        P(W>0) = P(X=1) = 1/2,since the probability of winning and losing 1 gamble is thesame. (b)   P(W<0) = P(X>2) = 1 - P(X=1) -P(X=2) = 1- 1/2 - (1/2)^2 = 1/4 (c)   X~Geom (1/2). So, E(X) = 1 / 0.5 = 2,based on expectation of a Geometric Distribution.        Then E(W) = 0,since W=0 when X=2.

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