Ten posts in a circle, exit at post 0 or post 6?

Question

Ten posts form a circle, numbered 0 to 9. Posts 0 and 6 are gates: landing on either ends the walk. A marker starts on post 2 and each step moves to an adjacent post, either direction with equal chance.

What is the probability the marker exits at post 0 rather than post 6?

Accepted Answer

The two gates split the loop into two arcs, and the marker starts on the arc 0, 1, 2, \dots, 6. It cannot cross a gate to reach the other arc (posts 7, 8, 9), so only this arc matters. On it, the walk is fair gambler's ruin on a line from 0 to 6, starting at 2. The chance of reaching the far end 6 first is 2/6 = 1/3, so the chance of exiting at 0 is 1 - \frac13 = \frac23. Takeaway. Reduce the ring to the arc between the gates: exit-at-0 probability is (distance to 6) over (length) = 4/6 = 2/3.

Ten posts in a circle, exit at post 0 or post 6?

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