Past exam of the mathematics course of the University of Cambridge / 2013 / ib / Paper 1 / 20H / ii / Solution

Let $T_i$ be the expected time to complete the specified directed transition, starting at $i$. This is a transition-completion time in a Markov chain, rather than merely a first visit to $c$. First-step decomposition givesThe missing $b\to c$ continuation in the middle equation represents successful completion, but its step is counted by the leading one. Substitution givesThus the requested expectation from $a$ is $128/11$ steps.