question about brownian hit time and reflexion principle I have a Brownian motion W(t)
I consider 2 events, where T is fixed :
A : W(T) is above a, a > 0
B : W(t) hit the level b, b < 0, at least once between 0 and T
I am trying to compute P(A and B)
This is comming from a book with several questions like this, with the hint being "use the reflexion principle"
Now, the reflexion principle helps me rephrashing either A or B, but doesn't help with the event (A and B ), as far as I can tell. I tried conditionning by either A or B but it doesn't reduce to something simple enough to be easily computed.
Does anyone know how to compute this probability  ?
thanks

Let $\tau_b$ denote the first hitting time of $b\lt0$ by $W$, then the process reflected at $b$ is defined by $X(t)=W(t)$ for $t\leqslant\tau_b$ and $X(t)=2b-W(t)$ for $t\gt\tau_b$ and the reflexion principle asserts that $X$ is again a Brownian motion. 
Then $A\cap B=[X(T)\lt2b-a]$ and, since $X(T)$ is distributed as $-\sqrt{T}\cdot W(1)$, one gets $$P(A\cap B)=P(W(1)\gt c)=\int_c^\infty\frac1{\sqrt{2\pi}}\mathrm e^{-x^2/2}\mathrm dx,\qquad c=\frac{a-2b}{\sqrt{T}}.$$
