Probability of a stopping time to be lower than another stopping time

Question

I was wondering on how to compute the probability of a stopping time being lower than another stopping time. More in details, just consider a drifted Brownian motion process $X_t$, and consider two thresholds $a,b$ such that $a<0<b$. Define now a stopping time $T_b=\min\{t\geq0|X^{max}_t\geq b\}$, where $X_t^{max}=\max_{\tau\in[0,t]}(X_{\tau})$ ($T_a$ similarly defined, with $X_t$ going beneath $b$). The first hitting time problem for a drifted Brownian motion is a well known result, but I was wondering what is the procedure to derive the probability measure $P(T_b<T_a\leq t)$. Putting it in words, this is the probability that the process hits first the upper bound and afterwards also the lower bound. I think that it can also be rewritten as $\int_0^t P\{T_b\leq\tau, T_a\in(\tau, t) \}d\tau$, but I do not find any useful hint on how to proceed with this way.