= Solution
By <conditional probability> and continuity of event times,
$$
\mathbb P(J=A\mid T<\tau)=\frac{F_A(\tau)}{F_A(\tau)+F_B(\tau)}.
$$
When $\tau>0$ and $q=\theta_A+\theta_B>0$, the common factor $1-e^{-q\tau}$ cancels, giving
$$
\boxed{\mathbb P(J=A\mid T<\tau)=\frac{\theta_A}{\theta_A+\theta_B}.}
$$
Before $\tau$, the constant <cause-specific hazards> divide the total event intensity in these proportions, regardless of how long the interval is. If $\tau=0$ or both intensities are zero, the conditioning event has <probability> zero and this <conditional probability> is undefined, not a ratio assigned by convention.
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