# $P(T_2=T_{-3}), P(T_1<T_4<T_{-1})$ and $P(T_3<2)-P(T_{-3}<2)$

Let $$W(t)$$ be a Brownian motion and $$T_x=\inf\{t:W(t)=x\}$$.

I need to calculate $$P(T_2=T_{-3}), P(T_1 and $$P(T_3<2)-P(T_{-3}<2)$$.

I'm not sure if I understand these correctly. I assume the first and the last one are $$0$$?

The second one I guess I can write as $$P(T_1, but I'm not really confident about this either.

Can anyone confirm if these are correct?

It follows directly from the definition of $$T_x$$ and the continuity of the sample paths of Brownian motion that $$T_1 almost surely. Hence, $$\mathbb{P}(T_1 Applying the formula $$\mathbb{P}(T_x (...which is a consequence of the fact that $$\mathbb{E}(W_{\tau})=0$$ for $$\tau:=\inf\{T_x,T_y\}$$...) we get $$\mathbb{P}(T_1