Simple Symmetric Random Walk on the Integer Values

Question

I want to prove that the simple symmetric random walk on the integer values with start in 0 is visiting the state N with probability 1/N+1 before visiting the negative numbers.

I already tried to do it by induction but it didn't work out for me.

Answer 1

HINT

The idea is to notice your random walk $(X_n)_{n=0}^\infty$ is a martingale and apply the Optional Stopping Theorem, which implies that is $n$ is the time when either $-1$ is hit or when $N>0$ is hit, then $$ \mathbb{E}[X_n] = \mathbb{E}[X_0] = \mathbb{E}[0] = 0. $$ Now note that $X_n \in \{-1,N\}$ by definition, so if $p$ is the probability $X_n = -1$ then $$ \mathbb{E}[X_n] = p(-1) + (1-p)N = N -p(N+1), $$ so you get $$ 0 = N -p(N+1). $$ Can you solve for $p$ and complete the problem?