Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 207 / 6 / i / Solution

Condition on the event time $T_A=u$. Independence makes the chance that $T_B$ is still later equal to its survivor function $F_B(u)$. HenceThe continuous distributions give zero probability of a tie. The complementary probability is ${\pi }_B={\int }_0^{\infty }{F}_A(u)f_B(u)du=1-{\pi }_A$.