Conditional probability of hitting time when drifted Brownian motion hits a barrier

Question

Let $a<0<b$ and $\mu\in \mathbb{R}$, and $X_t= \mu t+ B_t$ be a drifted brownian motion.

Is it possible to compute the following probability $$ \mathbb{P}[{\tau_a<\tau_b}\mid \tau_a], $$ where $\tau_a=\inf\{t\ge 0|X_t=a\}$ and $\tau_b=\inf\{t\ge 0|X_t=b\}$ are the first hitting times?