= Solution
The remaining arrivals occur in the next stop phase: $R\leq t+\tau_0\leq R+t$, or $R-t\leq\tau_0\leq R$. B waits until $R+r$, while A departed at $r$, giving $\tau_1=R\geq\tau_0$. This interval has length $t$, hence
$$
\boxed{\mathbb P(\tau_1\geq\tau_0\mid t)=\frac tR.}
$$
Endpoint assignments do not affect <probabilities>. The three outcomes are exhaustive and satisfy
$$
\boxed{\frac{r-t}{R}+\frac{R-r}{R}+\frac tR=1.}
$$
These are <conditional probabilities> for the stated fixed $t$; there is no additional averaging over A's arrival phase in the question.
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