Past exam of the options I course of the University of Oxford / 2009 / M / M2 / c / ii / Solution

State one is left after one step and never revisited. The eventual probability mass in the closed class $\{4,5\}$ isWithin this class the stationary distribution is $(b/(a+b),a/(a+b))$, because the balance equation is $a{\pi }_4=b{\pi }_5$. The two-state recurrence shows convergence from either starting state, since $|1-a-b|<1$. The other closed class never reaches state five. Therefore