Expected rounds for a fair chase to end

Question

A marker starts on position 1 of a short board numbered 0 to 5. Each round it moves forward one with probability 	frac12 or back one with probability 	frac12, stopping when it reaches position 0 or position 5. What is the expected number of rounds until the marker stops?

Accepted Answer

Let ek be the expected number of rounds from position k, with e0 = e5 = 0. First-step analysis gives ek = 1 + 	frac12 e{k-1} + 	frac12 e{k+1}, solved by the parabola ek = k(5 - k). At k = 1, e1 = 1 \cdot 4 = 4. Takeaway. Even one step from a wall, a fair walk averages k(N-k) rounds to finish; here 1 \cdot 4 = 4, because it often wanders away before being absorbed.

Expected rounds for a fair chase to end

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