 Solution

Let $h_k$ be the probability that the simple random walk started at $k$ hits $n$ before zero. The first-step analysis recurrence and boundary conditions areIf $p\ne q$, the characteristic equation of a linear recurrence has roots $1$ and $q/p$, so the boundary conditions giveIf $p=q=1/2$, the repeated root gives the limiting formula $\boxed{h_m=m/n}$. This is the gambler's ruin probability.

For a fixed integer stake $a\le N$, the fact that $a$ divides $N!$ lets us measure Patricia's wealth in units of £$a$. She starts at $M=N!/a$ and succeeds upon reaching $2M$. The formula above factors asFor $p=18/37$, $r=19/18>1$, so this probability increases as $M$ decreases. She should therefore choose the largest stake, $\boxed{a=N}$. For $p=19/37$, $r=18/19<1$, so the probability increases as $M$ increases; she should choose the smallest stake, $\boxed{a=1}$.