Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 3 / 8H / Solution

The communicating classes are $\{1,2,3,6\}$ and $\{4,5\}$; only the latter is closed. Let $h_i=P_i(T_2<\infty )$, with $h_2=1$ and $h_4=h_5=0$. The first-step equations give $h_3=1/2$, $h_6=(h_1+h_3)/2$, and $h_1=1/3+h_3/3+h_6/6$, hence $h_1=13/22$.