Harmonic hitting probability for the balanced-difference urn
= Harmonic hitting probability for the balanced-difference urn
Suppose an urn starts with $m$ green balls and $m+2$ red balls. Drawing green removes one green ball, while drawing red adds one green and one red ball. The difference remains two, and $h(g)=1/(g+1)$ is harmonic for the green-ball chain. Stopping $h$ on hitting $0$ or $N$ and then letting $N$ tend to infinity gives termination probability
$$
\mathbb P_m(\tau_0<\infty)=\frac1{m+1}.
$$