 Solution

Multiply survival to $t$ by the conditional type-A event probability in the next infinitesimal interval. The unconditional type-A increment probability isEquivalently, 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.