= Solution
Let $T=\min(T_A,T_B)$ be the first event time and $J$ its type. The <cause-specific hazard> for A is
$$
h_A(t)=\lim_{u\downarrow0}\frac{\Pr(t\leq T<t+u,\ J=A\mid T\geq t)}u.
$$
It is a rate among people free of both event types, not the marginal probability of eventually experiencing A. Under the given <independent> latent event times, their constant <hazard functions> give exponential <survivor functions>. Therefore \b[the first-event survivor is]
$$
\boxed{S(t)=\Pr(T_A\geq t,T_B\geq t)
=e^{-\alpha t}e^{-\beta t}=e^{-(\alpha+\beta)t}.}
$$
The combined first-event rate is $s=\alpha+\beta>0$. Subsequent formulas concerning conditioning on type A require $\alpha>0$.
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