Compute $\mathbb{P}\{ W_t < 0 \, \, \text{for all} \, \, 1 < t < 2\}$ for a Brownian motion $(W_t)_{t \geq 0}$ [closed]

Let $(W_t)_{t \ge 0}$ be a standard one-dimensional Brownian motion. Calculate $$\mathbb{P}\{ W_t < 0 \, \, \text{for all} \, \, 1 < t < 2\}.$$ I can only think that this will be conditional on $W_1$. Please tell how to proceed?

Hints: Set $$B_t := W_{1+t}-W_1.$$

1. Check (or recall) that $(B_t)_{t \geq 0}$ is a Brownian motion and that $(B_t)_{t \geq 0}$ is independent from $\mathcal{F}_1^W:=\sigma(W_s; s \leq 1)$.
2. Use the independence of $(B_t)_{t \geq 0}$ and $W_1$ to show that $$\mathbb{P}(W_t < 0 \, \, \text{for all} \, \, t \in (1,2) \mid W_1) = f(W_1)$$ where $$f(x) := \mathbb{P}(x+B_t < 0\, \, \text{for all $t \in (0,1)$}). \tag{1}$$
3. Prove that $f(x)=0$ for all $x \geq 0$. (Hint: What happens close to $t=0$?)
4. Fix $x<0$. Show that $$f(x) = \mathbb{P}(\tau_{-x} \geq 1)$$ for the stopping time $$\tau_{-x} := \inf\{t>0; B_t \geq -x\}.$$ Conclude from the reflection principle that $$f(x) = \mathbb{P}(|B_1|<-x),$$ and so $$f(x)=1-2\Phi(x)$$ where $\Phi$ is the cdf of the centered standard Gaussian distribution with density $\varphi$.
5. Combining the above steps gives \begin{align*} \mathbb{P}(W_t < 0 \, \, \text{for all} \, \, t \in (1,2)) &= \mathbb{E} \left[ \left( 1- 2\Phi(W_1)\right) 1_{\{W_1<0\}} \right] \\ &= \frac{1}{2} - 2 \int_{-\infty}^0 \Phi(x) \varphi(x) \, dx. \end{align*}
6. Conclude that $$\mathbb{P}(W_t < 0 \, \, \text{for all} \, \, t \in (1,2)) = \frac{1}{4}; $$ see e.g. this answer for how to compute the integral in Step 5.