This problem cropped up in some research I am doing. I imagine it is standard, but I cannot seem to find the answer.
Let $W_t$ be a standard Brownian motion. Suppose there are four values $a < 0 < b$ and $c<d$. For a given $t > 0$, I want to know the probability that $W_t$ reaches $b$ before reaching $a$ and then after reaching $b$ never falls below $c$ nor exceeds $d$ until time $t$.