A symmetric random walk on a finite line from 0 to up to x

Question

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.

Answer 1

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

$$ 2^{-n}\sum_{k=0}^n\binom nk\mathbb1_T(x_0+n-2k)\;. $$