  We present the universal features of the hitting probability $Q(x,L)$, the
probability that a generic stochastic process starting at $x$ and evolving in a
box $[0,L]$ hits the upper boundary $L$ before hitting the lower boundary at 0.
For a generic self-affine process (describing, for instance, the polymer
translocation through a nanopore) we show that $Q(x,L)=Q(x/L)$ and the scaling
function $Q(z)\sim z^\phi$ as $z\to 0$ with $\phi=\theta/H$ where $H$ and
$\theta$ are respectively the Hurst exponent and the persistence exponent of
the process. This result is verified in several exact calculations including
when the process represents the position of a particle diffusing in a
disordered potential. We also provide numerical supports for our analytical
results.
