  We study theoretically, experimentally and numerically the probability
distribution $F(t_f|x_0,L)$ of the first passage times $t_f$ needed by a freely
diffusing Brownian particle to reach a target at a distance $L$ from the
initial position $x_0$, taken from a normalized distribution $(1/\sigma)\,
g(x_0/\sigma)$ of finite width $\sigma$. We show the existence of a critical
value $b_c$ of the parameter $b=L/\sigma$, which determines the shape of
$F(t_f|x_0,L)$. For $b>b_c$ the distribution $F(t_f|x_0,L)$ has a maximum and a
minimum whereas for $b<b_c$ it is a monotonically decreasing function of $t_f$.
This dynamical phase transition is generated by the presence of two
characteristic times $\sigma^2/D$ and $L^2/D$, where $D$ is the diffusion
coefficient. The theoretical predictions are experimentally checked on a
Brownian bead whose free diffusion is initialized by an optical trap which
determines the initial distribution $g(x_0/\sigma)$. The presence of the phase
transition in 2d has also been numerically estimated using a Langevin dynamics.
