= Solution
Condition throughout on the given arrival time $t$ of A, with $0\leq t<r$. A departs at time $r$. B also departs at $r$ exactly when it arrives before that release, so $t+\tau_0\leq r$, or $0<\tau_0\leq r-t$. Since $\tau_0$ is uniform on an interval of length $R$,
$$
\boxed{\mathbb P(\tau_1=0\mid t)=\frac{r-t}{R}.}
$$
The original timeline below shows the three disjoint ranges of $\tau_0$. The same idealized simultaneous-release convention as in part (a) is used.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1-arrival-timeline.png]
{title=Conditional arrival ranges for the following vehicle at a fixed arrival phase of the leading vehicle}
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