Distribution of the argmax of Brownian motion.

Question

Let $B:\Omega\times[0,1]\to\mathbb R$ be a standard Brownian motion starting at zero defined on some probability space $(\Omega,\mathcal{F},\mathbb P)$. For any $a\in[0,1]$, I am interested in $$F(a):=\mathbb P\{\omega\in\Omega:\mathrm{argmax}_{t\in[0,1]}B_t(\omega)\leq a\}$$ This is the cdf of a random variable in $[0,1]$. Notice that this is well defined because the argmax is unique almost surely. (See Th2.11 of Mörters-Peres)