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

Condition throughout on the given arrival time $t$ of A, with $0\le t<r$. A departs at time $r$. B also departs at $r$ exactly when it arrives before that release, so $t+{\tau }_0\le r$, or $0<{\tau }_0\le r-t$. Since ${\tau }_0$ is uniform on an interval of length $R$,The original timeline below shows the three disjoint ranges of ${\tau }_0$. The same idealized simultaneous-release convention as in part (a) is used.

If B arrives during the remaining pass phase, then $r<t+{\tau }_0<R$, or $r-t<{\tau }_0<R-t$. It passes immediately, soBecause $r-t>0$, this is strictly between zero and ${\tau }_0$. The phase interval has length $(R-t)-(r-t)=R-r$, giving

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.