Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 3 / 9H (Markov Chains)

In the order $(1,2,3,L,G)$, where $L$ means left without a degree and $G$ means graduated, the transition matrix isHere $p+q+r=1$.

The three year states are transient states: from each there is positive probability of entering an absorbing state, and repeated trials make eventual departure certain because $q<1$. The singleton classes $\{L\}$ and $\{G\}$ are closed absorbing states, hence recurrent. Thus the chain has two closed recurrent classes and three transient singleton classes.

During any year, repeats may be ignored until the first non-repeat outcome. Conditional on that event, the chance of progressing isThe student must progress successfully through all three years. By the Markov property, the required probability is therefore