Suppose one has $1 now, and he wants to win another N dollars, where N ∼ Poisson(λ). He tosses a fair coin repeatedly; if it comes up heads, he wins one dollar, but if it comes up tails, he loses one dollar. He plays this game repeatedly until one of two events occurs: either he runs out of money and is bankrupted or he reaches his goal. What is the probability that he is ultimately won?

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