Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 5 / d / i / Solution

Write $q=P_{BI}$ and $p=P_{AI}=1-q$. A path with $n$ visits to $I$ makes $n-1$ unsuccessful excursions back to $B$, followed by one successful step to $A$. Every $B\to I$ step has probability one, soThe visit count has a geometric distribution, with success meaning the final $I\to A$ step.