Let $W$ denotes a Wiener process and $\gamma=\max\left\{ t\in\left[0,2\right]:W_{t}=0\right\}$ . What is the probability, that $$\mathbf{P}\left(\max_{s\in\left[0,\gamma\right]}W_{s}>\sqrt{\gamma}\right)?$$ I have some problems with the scaling, because I know some hints, I just can't put them together.
Hints:
1. If $\tau=\inf\left\{ t>1:W_{t}=0\right\}$ and $\nu=\max\left\{ t\in\left[0,1\right]:W_{t}=0\right\}$ , then $$\tau\overset{d}{=}\frac{1}{\nu}.$$
2. $\left(\frac{1}{\sqrt{\nu}}B_{t\cdot\nu}\right)_{t\in\left[0,1\right]}$ is a Brownian bridge, which is independent from $\nu$.
3. I think we should convert $\max_{s\in\left[0,\gamma\right]}W_{s}$ somehow to the Brownian bridge given above, and we could calculate the probability that for a given level the Brownian bridge reaches this level.