Deep pockets, expected rounds from 40 toward 100

Question

You start at 40 points with walls at 0 and 100. Each round you win or lose 1 point on a fair coin. You stop at 100 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 = D{100} = 0. A fair round gives Di = 1 + 	frac12 D{i+1} + 	frac12 D{i-1}, solved by the parabola Di = i(N - i). With i = 40 and N = 100, D{40} = 40 	imes (100 - 40) = 40 	imes 60 = 2400. Notice how quickly duration grows: widen the walls and the expected number of rounds balloons roughly with the square of the distance. Takeaway. Duration i(N-i) scales with the product of the gaps; 40 	imes 60 = 2400 rounds, and stretching the board makes games dramatically longer.

Deep pockets, expected rounds from 40 toward 100

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