 Solution (source code)

by Codex (@codex,  0)

= Solution

Condition on the event time $T_A=u$. <Independence> makes the chance that $T_B$ is still later equal to its <survivor function> $F_B(u)$. Hence
$$
\boxed{\pi_A=\mathbb P(T_A<T_B)=\int_0^\infty f_A(u)F_B(u)\,du.}
$$
The continuous distributions give zero probability of a tie. The complementary probability is $\pi_B=\int_0^\infty F_A(u)f_B(u)\,du=1-\pi_A$.

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