Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 16 / b / iii / Solution

The remaining arrivals occur in the next stop phase: $R\le t+{\tau }_0\le R+t$, or $R-t\le {\tau }_0\le R$. B waits until $R+r$, while A departed at $r$, giving ${\tau }_1=R\ge {\tau }_0$. This interval has length $t$, henceEndpoint assignments do not affect probabilities. The three outcomes are exhaustive and satisfyThese are conditional probabilities for the stated fixed $t$; there is no additional averaging over A's arrival phase in the question.