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

A path with $n$ visits makes $n$ departures from $B$ and $n$ departures from $I$. Its conditional mean duration is therefore $n({\tau }_B+{\tau }_I)$. Averaging over the geometric distribution,Only the mean waiting times are needed here: linearity of expectation adds their contributions without requiring each residence time to equal its mean.