Steps for a wanderer to reach home or the bar

Question

A wanderer is 2 blocks from home (position 0) and 3 blocks from the bar (position 5). Each minute he steps one block toward home or one block toward the bar, each with probability 1/2. He stops when he reaches either home or the bar. On average, how many steps does that take?

Accepted Answer

His position is a symmetric random walk on 0, 1, \ldots, 5 absorbed at both ends, starting at i = 2 (since he is 2 blocks from home). The expected remaining steps from position i solves Di = 1 + 	frac12 D{i-1} + 	frac12 D{i+1}, \quad D0 = D5 = 0, with solution Di = i(N - i), here N = 5: D2 = 2 \cdot (5 - 2) = 2 \cdot 3 = 6 	ext{ steps.} Takeaway. Expected duration of a fair \pm 1 walk on 0..N from i is i(N-i); two blocks from home with the bar five away gives 2 \cdot 3 = 6.

Steps for a wanderer to reach home or the bar

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