= Solution
Multiply survival to $t$ by the conditional type-A event probability in the next infinitesimal interval. \b[The unconditional type-A increment probability is]
$$
\boxed{\Pr(t\leq T<t+dt,J=A)=e^{-(\alpha+\beta)t}\alpha\,dt+o(dt).}
$$
Equivalently, its exact value over a positive interval of length $u$ is $\int_t^{t+u}\alpha e^{-(\alpha+\beta)v}dv$. The factor $e^{-(\alpha+\beta)t}$ is essential because a prior B event removes an individual from the relevant <risk set>.
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