Average moves for a beetle to leave a strip

Question

A beetle stands on cell 2 of a strip of cells numbered 0 to 6. Each step it walks to an adjacent cell, right or left with equal probability 	frac12. It leaves the strip when it reaches cell 0 or cell 6. On average, how many steps does the beetle take before leaving?

Accepted Answer

Let ek be the expected number of steps from cell k, with e0 = e6 = 0. First-step analysis gives ek = 1 + 	frac12 e{k-1} + 	frac12 e{k+1}. The "+1" per step bends the solution into a parabola through the two zero endpoints: ek = k(6 - k), \qquad e2 = 2 \cdot 4 = 8. Takeaway. Expected absorption time for a fair walk between 0 and N is k(N-k); standing at cell 2 of a length-6 strip gives 8 steps on average.

Average moves for a beetle to leave a strip

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