Brownian motion above zero at both t = 2 and t = 8

Let $B$ be a standard Brownian motion. What is the probability that $B_2$ and $B_8$ are both strictly positive?

$1/3$. For $s<t$ the correlation of $B_s$ and $B_t$ is $\rho=\sqrt{s/t}=\tfrac12$. For a standard bivariate normal, $\mathbb{P}[X>0,Y>0]=\tfrac14+\tfrac{\arcsin\rho}{2\pi}=\tfrac14+\tfrac{1}{12}=\tfrac13$.