Expected duration of biased gambler's ruin ($m_i=[N(1-{\rho }^i)/(1-{\rho }^N)-i]/(p-q)$)

For a nearest-neighbour fortune on $\{0,\dots ,N\}$, with win probability $p$, loss probability $q=1-p$ and $p\ne q$, put $\rho =q/p$. First-step analysis gives $m_i=1+p{m}_{i+1}+q{m}_{i-1}$, with zero endpoint values. Solving the recurrence gives the displayed mean duration. Its symmetric limit is $i(N-i)$.