  We consider a percolation model on square lattices with sites weighted by
beta-distributed random variables $S\sim \mathrm{Beta}(a,b)$ with a positive
real parameters $a>0$ and $b>0$. Using the Monte Carlo method, we estimate the
percolation probability $P_\infty$ as a relative frequency $P^*_\infty$
averaged over the target subset of sites on a square lattice. As a result of
the comparative analysis, we formulate two empirical hypotheses: the first on
the correspondence of percolation thresholds $p_c$ to $p_0$-quantiles (where
$p_0=0.592746\ldots$) of random variables $S_i$ weighing sites of the square
lattice with $(1,0)$-neighborhood, and the second on the convergence of
statistical estimates of percolation probability functions $P^*_\infty(p)$ to
cumulative distribution functions $F_{S_i}(p)$ of these variables $S_i$ for the
supercritical values of the occupation probability $p\geq p_c$.
