Let $T > 0$ be a fixed point in time, e.g. $T=1$.
Consider the stochastic process $B_t = (T-t) \cdot W_t + t \cdot V_{T-t}$, where $V_t \sim W_t$ for all $t \geq 0$.
We want to show that $(B_t)_{t \geq 0}$ is a Brownian Bridge, i.e. we need to calculate $\mathbb{P}(B_0 = 0)$ and $\mathbb{P}(B_1 = 0)$ to show that the end value is equal to the start value.
How do I determine $\mathbb{P}(B_0 = 0)$ and $\mathbb{P}(B_1 = 0)$?
My approach: w calculate $\mathbb{P}(B_0 = 0) = \mathbb{P}(W_0 = 0) = 1$ since $(1-0) \cdot W_t + 0 \cdot V_{1-0} = W_t$. Then we show $\mathbb{P}(B_1 = 0) = \mathbb{P}(V_0 = 0) = \mathbb{P}(W_0 = 0) = 1$ since $(1-1) \cdot W_t + 1 \cdot V_{1-1} = V_0$ and $V_t \sim W_t$.