Simulation statistical modeling in the study of the laws of large numbers

The paper considers the theoretical foundations of the Monte Carlo method based on the limit theorems of probability theory. Simulations of a coin toss and the number of determination are given and described. The authors proved experimentally that sufficiently high accuracy of solving problems with the Monte Carlo method application is guaranteed only when a large number of experiments are carried out. Therefore, the Monte Carlo method is practically implemented only with the computer. The procedure for generating random numbers is a fundamental basis of the statistical test method. The average values tend to some constant with an unlimited increase in the number of experiments. It was found on the basis of a comparative analysis of the obtained results when studying the laws of large numbers by the method of simulation statistical modeling in MathCad and MS Excel. The Limit theorems guarantee the high quality of statistical estimates obtained by the Monte Carlo method for a large model dimension n. However, practically acceptable estimates of the simulated system’s characteristics can often be obtained even at relatively small values of n (of a few tens and hundreds). It has been proven that averaging the values obtained over several runs of the model significantly improves the quality of statistical estimates.