 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-207/6/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

Condition on the event time

T_{A} = u

. Independence makes the chance that

T_{B}

is still later equal to its survivor function

F_{B} (u)

. Hence

π_{A} = P (T_{A} < T_{B}) = \int_{0}^{\infty} f_{A} (u) F_{B} (u) d u .

The continuous distributions give zero probability of a tie. The complementary probability is

π_{B} = \int_{0}^{\infty} F_{A} (u) f_{B} (u) d u = 1 - π_{A}

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