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A known problem with a twist…

The following was sent as an email to me. It’s the old fashioned gablre’s ruin problem with one more option. If you love theory then this is a good treat 😉

A gambler plays the following game. He starts with r dollars, and is trying to end up with \alpha dollars. At each go he chooses an integer s between 1 and min(r,\alpha-r) and then tosses a fair coin. If the coin comes up heads, then he wins s dollars, and if it comes up tails then he loses s dollars.

The game finishes if he runs out of money (in which case he loses) or reaches \alpha dollars (in which case he wins). Prove that whatever strategy the gambler adopts (that is, however he chooses each stake based on what has happened up to that point), the probability that the game finishes is 1 and the probability that the gambler wins is r/\alpha .

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