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

Each visit to $I$ corresponds to two steps, one out of $B$ and one out of $I$. Hence a successful path with $n$ visits contains $2n$ steps. Taking its expectation over the geometric distribution givesThe last pair is $B\to I\to A$; it contributes two steps just like each previous $B\to I\to B$ excursion.

Using the differentiated geometric series,This is finite for $p>0$, equals two when every visit succeeds, and diverges as the chance of success tends to zero.