Random Walk on Nonnegative Integers
A particle performs a random walk on the non-negative integers as follows. When at the point n (> 0) its next position is uniformly distributed on the set {0, 1, 2,..., n + 1}. When it hits 0 for the first time, it is absorbed. Suppose it starts at the point a. Find the probability that its position never exceeds a.
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Such advanced question warrants a bounty of 25$ I believe