Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 7H (Markov Chains)

With states ordered as $(A,B,C)$ and rows representing the current state, the Markov chain has transition matrix

Starting from the airport, after one step the distribution is $(0,1/2,1/2)$. Multiplying once more by $P$ gives

Let $p_n=\mathbb{P}(X_n=A)$. Whenever the taxi is away from the airport it returns with probability $3/4$, whereas it must leave whenever it is at the airport. HenceSubtracting the fixed point $3/7$ givesThereforefor every $n\ge 0$, and in particular for the requested $n\ge 1$.