Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 2 / 18H / Solution

Invariant means $\pi P=\pi$. In a finite irreducible chain, $E_i{T}_i^{+}=1/{\pi }_i$ and the expected visits to $i$ between visits to $j$ are ${\pi }_i/{\pi }_j$. The first burglar has uniform invariant law on $64$ PINs, so (a) is $64$ and (b) is $16$. First-step equations by Hamming distance give the adjacent-state hitting time (c) as $63$. For the modified one-coordinate chain, destination weights are $1$ for digits $0$–$5$ and $2$ for $6,7$; its invariant weights are proportional to $w_a(10-w_a)$, namely $9$ and $16$. The product invariant law and the occupation formula give (d) ${\pi }_{(6,7)}/{\pi }_{(0,0)}=256/81$.

Solved by gpt-5.6-sol high.