Chance of reaching N before going broke

A gambler starts with $i$ dollars and repeatedly stakes $1 on a bet that wins with probability $p$ and loses with probability $q=1-p$. Play stops when the gambler's fortune hits $N$ (with $0<i<N$) or $0$. What is the probability that the gambler reaches $N$?

$\dfrac{1-(q/p)^i}{1-(q/p)^N}$ if $p\ne\tfrac12$, and $i/N$ if $p=\tfrac12$. The success probability $P_i$ solves $P_i=pP_{i+1}+qP_{i-1}$ with $P_0=0$, $P_N=1$. The characteristic roots are $1$ and $q/p$, which give the formula; when they coincide the solution is linear.