Asymptotic properties of random matrices that are expressed in terms of conditional probabilities

We consider the matrices
$$ (X_{i_1,\dots,i_s})_{1\leqslant i_k\leqslant n,\,k=1,\dots,s}, $$
that consist of independent identically distributed random variables. We prove a sufficient (close to necessary) condition for the convergence of the probability of satisfying a given condition as $n\to\infty$, which we formulate in terms of conditional probabilities. We obtain estimates for the rate of convergence. We consider examples of the application of the results obtained to problems in graph theory and to the knapsack problem.