Expected moves before a lift reaches an end floor

Question

A test lift starts on floor 5 of a shaft with floors numbered 0 to 12. Each move it goes up one floor with probability 	frac12 or down one with probability 	frac12. The test ends when it reaches floor 0 or floor 12. What is the expected number of moves until the test ends?

Accepted Answer

Let ek be the expected number of moves from floor k, with e0 = e{12} = 0. First-step analysis over the fair move gives ek = 1 + 	frac12 e{k-1} + 	frac12 e{k+1}, whose solution through the zero endpoints is the parabola ek = k(12 - k). At k = 5, e5 = 5 \cdot 7 = 35. Takeaway. The mean time for a symmetric walk to reach either end is k(N-k); five floors up a twelve-floor shaft gives 35 moves on average.

Expected moves before a lift reaches an end floor

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