Does Brownian motion go positive immediately?

Let $W$ be a standard Brownian motion and $M_t=\sup_{0\le s\le t}W_s$ its running maximum. For an arbitrary $t>0$, what is $\mathbb{P}[M_t>0]$?

$1$. By the reflection principle $M_t$ has the law of $|W_t|$, which is positive almost surely. Alternatively, the event that $W$ is positive at arbitrarily small times has positive probability and is in the germ field, so Blumenthal's 0-1 law forces probability $1$.