How many fair bets until the game ends?

Question

You start at 20 points and bet 1 point on a fair coin each round, winning or losing 1 point with probability $1/2$ each. You stop when you reach 50 points or drop to 0 points.

On average, how many rounds until the game ends?

Accepted Answer

Let Di be the expected number of rounds to finish, starting from i, with D0 = DN = 0 (already stopped). One round takes you to i \pm 1, so Di = 1 + 	frac12 D{i+1} + 	frac12 D{i-1}. The "+1" counts the round just played. The function that fits both boundaries is the parabola Di = i(N - i), which is the distance to the bottom times the distance to the top. Here i = 20, N = 50: D{20} = 20 	imes (50 - 20) = 20 	imes 30 = 600. Takeaway. A fair game lasts i(N-i) rounds on average, distance to one wall times distance to the other; it runs longest when you start in the middle.

How many fair bets until the game ends?

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