= Solution
Divide the joint first-event/type-A density by the type probability. For $\alpha>0$,
$$
f_{T\mid J=A}(t)=\frac{\alpha e^{-(\alpha+\beta)t}}{\alpha/(\alpha+\beta)}
=\boxed{(\alpha+\beta)e^{-(\alpha+\beta)t},\qquad t\geq0.}
$$
It integrates to one, so \b[the conditional time is exponential with rate] $\alpha+\beta$. Type-A events that win the competition tend to occur sooner than an unrestricted latent A time. In fact the same conditional density holds for B, so the observed first-event time and its type are <independent> in this constant-rate model.
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