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

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.

The same geometric series identity givesA direct renewal check gives the identical result: $T_B={\tau }_B+T_I$ and $T_I={\tau }_I+q{T}_B$, so $p{T}_B={\tau }_B+{\tau }_I$. This also checks the units: mean first-passage time has units of time, unlike the dimensionless step count.