Counting paths that touched a high before finishing

Question

A stock does a symmetric \pm 1 tick walk for 20 steps. Consider only the paths that finish exactly +8 ticks from where they started. Among those paths, what fraction reached +10 (or higher) at some point before finishing?

Accepted Answer

Fix the endpoint at +8 after 20 steps. A path ending at +8 has u up-steps and d down-steps with u - d = 8 and u + d = 20, so u = 14: there are \binom{20}{14} = \binom{20}{6} = 38{,}760 such paths. Reflect at the barrier +10. Take any path that ends at +8 and touches +10. After the first moment it hits +10, reflect the remainder across the line y = 10: every up becomes a down and vice versa. A path that ended at +8 (which is 2 below the barrier) now ends at 10 + 2 = +12. This reflection is a bijection between - paths ending at +8 that touch +10, and - all paths ending at +12 (every such path l…

Counting paths that touched a high before finishing

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