 Solution

ID: 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/ii/solution

If B arrives during the remaining pass phase, then

r < t + τ_{0} < R

, or

r - t < τ_{0} < R - t

. It passes immediately, so

τ_{1} = (t + τ_{0}) - r = τ_{0} - (r - t) .

Because

r - t > 0

, this is strictly between zero and

τ_{0}

. The phase interval has length

(R - t) - (r - t) = R - r

P (0 < τ_{1} < τ_{0} ∣ t) = \frac{R - r}{R} .

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