Probability and the concept of randomness

After A. N. Kolmogorov, it is customary to distinguish between a proper randomness as the absence of regularity and a stochastic randomness as the subject of probability theory. On the example of the binary sequences (0110100...) we give a presentation of the main approaches to the definition of the “individual random sequences”. We focus on the von Mises approach based on the notion of “Kollektiv”, on the algorithmic approach of Church and Kolmogorov, on the typical approach of Martin-Löf, on the complexity approach of Kolmogorov, and on the nonpredictable approaches of Ville and Uspensky. In Introduction, we give a brief survey of the main periods in the formation of probability theory. The talk is timed to the 75th anniversary of the publication of “Foundation of the Theory of Probability” by A. N. Kolmogorov.