= Solution
An individual has neither died nor been censored by time $t$ exactly when $T>t$ and $C>t$. Under <independent censoring>, this has probability $F_T(t)G(t)$. Thus \b[the probability of leaving observation by either route] is
$$
\boxed{\mathbb P(\min(T,C)\le t)=1-F_T(t)G(t)=\begin{cases}1-F_T(t),&0\le t<\tau_c-\tau_b,\\1-F_T(t)\dfrac{\tau_c-\tau_a-t}{\tau_b-\tau_a},&\tau_c-\tau_b\le t<\tau_c-\tau_a.\end{cases}}
$$
The two pieces agree at $t=\tau_c-\tau_b$. The expression counts the first of death and censoring, so it does not double-count individuals who would eventually experience both.
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