A symmetric random walk ( probability of moving left by 1 = probability of moving right by 1) on a subset of the integers from $0$ up to $x$. Take some $y$ s.t. $0 \leq y < x$ and let the random walk be considered a success if starting at some $y < x_0 \leq x$ we are in the '$0$ up to $y$ subset' after $n$ steps. What is the probability of success after $n$ steps starting at $x_0$?
edit : At the boundary the walker is guaranteed to move back inwards for its next step. It simply bounces back.
Best Answer
Since the walk is reflected at the boundaries, it’s equivalent to an unbounded random walk in which the target for success is $T=[-y,y]+2x\mathbb Z$. In this unbounded walk, the position after $n$ steps is binomially distributed, so the probability for success is
$$ 2^{-n}\sum_{k=0}^n\binom nk\mathbb1_T(x_0+n-2k)\;. $$