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Here is the idea. Let the vertices be $v_0,\ldots,v_n$, where $v_0$ is the origin and the vertices are arranged in clockwise order. The first step is either clockwise or counterclockwise — let's say clockwise. Then the probability you're after is the probability that you reach $v_n$ before you reach $v_0$, starting at $v_1$. This is random walk on a line, so ...