Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 28J / a / Solution

Superpose the two independent Poisson processes. The combined process has rate $9$, and each arrival is independently a Number 1 bus with probability $1/9$ and a Number 8 bus with probability $8/9$. Therefore the number of Number 8 buses before the first Number 1 has the geometric distribution on $\{0,1,2,\dots \}$, andThis is number of one Poisson type before another with rates $1$ and $8$.