On tables of random numbers

Abstract

The set theoretic axioms of the calculus of probability, in formulating which I had the opportunity of playing some part (Kolmogorov, 1950), had solved the majority of formal difficulties in the construction of a mathematical apparatus which is useful for a very large number of applications of probabilistic methods, so successfully that the problem of finding the basis of real applications of the results of the mathematical theory of probability became rather secondary to many investigators. I have already expressed the view [see Kolmogorov (1950), Chapter I] that the basis for the applicability of the results of the mathematical theory of probabi lity to real 'random phenomena' must depend on some form of the frequency concept of probability, the unavoidable nature of which has been established by von Mises in a spirited manner. However, for a long time I had the following views. (1) The frequency concept based on the notion of limiting frequency as the number of trials increases to infinity, does not contribute anything to substantiate the applicability of the results of probability theory to real practical problems where we have always to deal with a finite number of trials. (2) The frequency concept applied to a large but finite number of trials does not admit a rigorous formal exposition within the framework of pure mathematics. Accordingly I have sometimes put forward the frequency concept which involves the conscious use of certain not rigorously formal ideas about 'practical reliability', 'approximate stability of the frequency in a long series of trials', without the precise definition of the series which are-'sufficiently large' etc. [see Foundation^ of the Theory of Probability, Chapter I and for more details Great Soviet Encyclo paedia (section on Probability) and Mathematika iou metod i Znachenye (Chapter on Probability Theory)]. I still maintain the first of the two theses mentioned above. As regards the second, however, I have come to realise that the concept of random distribution of a property in a large finite population can have a strict formal mathematical exposition. In fact, we can show that in sufficiently large populations the distribution of the pro perty may be such that the frequency of its occurrence will be almost the same for all sufficiently large sub-populations, when the law of choosing these is sufficiently simple. Such a conception in its full development requires the introduction of a measure of the complexity of the algorithm. I propose to discuss this question in another article. In the present article, however, I shall use the fact that there cannot be a very large number of simple algorithms. 369 3