Drifted Brownian interval-exit probability ($p_{-}(x)=\frac{e^{-2\mu x}-e^{-2\mu b}}{e^{2\mu b}-e^{-2\mu b}}$)

For Brownian motion with drift $dX=dB+\mu dt$ in $(-b,b)$, the displayed probability of exiting at $-b$ follows by stopping the harmonic function $e^{-2\mu X}$. A unit-time increment larger than $2b$ has a uniform positive probability and forces exit, so the exit time has a geometric tail bound from a uniform escape probability and all polynomial moments. At zero drift the limiting probability is $(b-x)/(2b)$.