Short fair game, expected rounds from 5 up to 8

Question

You start at 5 points with walls at 0 and 8. Each round you win or lose 1 point on a fair coin. You stop at 8 points or at 0 points. On average, how many rounds until the game ends?

Accepted Answer

Let Di be the expected number of rounds from i, with D0 = D8 = 0. A fair round gives Di = 1 + 	frac12 D{i+1} + 	frac12 D{i-1}, solved by Di = i(N - i). With i = 5 and N = 8, D5 = 5 	imes (8 - 5) = 5 	imes 3 = 15. Distance 5 to the floor and 3 to the ceiling multiply to 15 expected rounds. Takeaway. Even a short fair game follows i(N-i): 5 	imes 3 = 15 rounds on average.

Short fair game, expected rounds from 5 up to 8

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