Conditional mean of a Brownian bridge at t = 1/2

Build a Brownian bridge on $[0,1]$ from a standard Brownian motion $W$ via $X_t=W_t-tW_1$. Given that $X_{3/4}=3$, what is the expected value of $X_{1/2}$?

$2$. For $s<t$, $\operatorname{Cov}(X_s,X_t)=s(1-t)$ and $\operatorname{Var}(X_t)=t(1-t)$, so the Gaussian regression gives $\mathbb{E}[X_s\mid X_t]=\tfrac{s}{t}X_t=\tfrac{1/2}{3/4}\cdot3=2$.