Gambler's Ruin Problem

The Problem

A gambler starts with \iand plays a game where, at each step, they win $1 with probabilitypand lose $1 with probabilityq = 1 - p$.Thegameendswhenthegamblereitherreaches$$N$ (win) or $0 (ruin).

Let $P_i$ denote the probability of reaching \Nbefore $0, starting from$i$.

1. Derive a closed-form expression for $P_i$ in terms of $p$, $q$, $i$, $N$.
2. What happens in the fair case $p=q=\frac{1}{2}$?
3. What is the expected duration of the game (number of steps until absorption)?