Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 1 / 20H / b / Solution

Let $T_k$ be the expected capture time starting from $D_0=k$, so $T_0=0$. The Strong Markov property at successive first passages to lower levels and part (a) giveAt $D_n=1$, the next state is zero with probability $r/2$, one with probability $1/2$, and two with probability $(1-r)/2$. First-step analysis therefore givesorSolving,Substituting $\mu =2/(2q-1)$ now yieldswhich is the required expected capture time.