Symmetric walk hitting 10 before 0 from 6

A token moves on the integers $0,1,\dots,10$. At each step it goes up or down by one with equal probability, and it stops for good on reaching either $0$ or $10$. Starting at $6$, what is the probability it stops at $10$?

$\tfrac35$. The position of a symmetric walk is a martingale, so by optional stopping $6=10\cdot p+0\cdot(1-p)$, giving $p=0.6$. Equivalently, for gambler's ruin the hit probability is linear in the start point.