Balanced start, rounds to finish from the midpoint

Question

You start at 10 points, exactly halfway between 0 and 20. Each round you win or lose 1 point on a fair coin. You stop at 20 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{20} = 0. One fair round gives Di = 1 + 	frac12 D{i+1} + 	frac12 D{i-1}, whose boundary-fitting solution is the parabola Di = i(N - i). With i = 10 and N = 20, D{10} = 10 	imes (20 - 10) = 10 	imes 10 = 100. Starting at the midpoint gives the longest possible game for these walls, since i(N-i) peaks at the center. Takeaway. Duration is i(N-i), largest at the midpoint; here 10 	imes 10 = 100 rounds.

Balanced start, rounds to finish from the midpoint

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