The optimal estimates of the probability of a random event

Abstract

One of the most important problems of designing production systems is to ensure their reliability. At the same time, it is meant not only the technical reliability of technological equipment, but also the influence of external and internal random factors that lead to failures of the production process. In most works related to the study of regularities of random processes in production and other technical systems, the known axiomatics is used. The corresponding analytical tool allows to solve the problem of assessing the probability of production tasks (for example, daily schedule of steel smelting or monthly production plan). The known formula for estimating the probability of some random event, which determines the specified probability as the ratio of the number of successful experiments to the number of all experiments, is usually accepted as an axiom. There is no doubt that this axiom is fair, as it is always confirmed by experience. However, it is interesting to obtain this confirmation analytically. This paper presents an analytical conclusion of this formula. According to the frequency axiomatics of probability theory, the probability of some event p* is determined by the ratio of the number of realizations of this event y in a series of n independent tests to the number of these tests. The value of y has a binomial distribution, which at sufficiently large n by Laplace’s theorem tends to normal with the same parameters. Almost normal law can be used already at n > 15 . We write the evaluation model p* as the relative frequency, that is, the ratio y to n. The distribution law of p* as a linear function of y is also asymptotically normal. The numerical characteristics of the distribution of p* are calculated by known formulas for linear functions of random variables. Questions arise: is the estimate of p* optimal and how to estimate its variance to then compute a*[p*]? To answer these questions, an optimization problem is formulated, for which the Lagrange multiplier method is used. The solution of this problem has shown that the axiomatic values of the 1/n multipliers used are optimal in the sense of the minimum variance of the random event probability estimate.