= Solution
Take $s$ on the horizontal axis and $t$ on the vertical axis, both nonnegative. There is no interruption when the noise occurs after the scheduled ending, namely \b[$t>s$]. This is the part of the first quadrant above the diagonal $t=s$. Equality has <probability> zero, so assigning the boundary either way does not affect the calculation.
The same original diagram below also marks the region used in part (e); its lower boundary offset is an illustrative positive choice of $r_0$.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1-exponential-regions.png]
{title=The no-interruption region above the diagonal and the region of positive remaining time exceeding a chosen threshold}
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