  The universality of the crossing probability $\pi_{hs}$ of a system to
percolate only in the horizontal direction, was investigated numerically by
using a cluster Monte-Carlo algorithm for the $q$-state Potts model for
$q=2,3,4$ and for percolation $q=1$. We check the percolation through
Fortuin-Kasteleyn clusters near the critical point on the square lattice by
using representation of the Potts model as the correlated site-bond percolation
model. It was shown that probability of a system to percolate only in the
horizontal direction $\pi_{hs}$ has universal form $\pi_{hs}=A(q) Q(z)$ for
$q=1,2,3,4$ as a function of the scaling variable $z= [
b(q)L^{\frac{1}{\nu(q)}}(p-p_{c}(q,L)) ]^{\zeta(q)}$. Here,
  $p=1-\exp(-\beta)$ is the probability of a bond to be closed, $A(q)$ is the
nonuniversal crossing amplitude, $b(q)$ is the nonuniversal metric factor,
$\zeta(q)$ is the nonuniversal scaling index, $\nu(q)$ is the correlation
length index.
  The universal function $Q(x) \simeq \exp(-z)$. Nonuniversal scaling factors
were found numerically.
