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The notion of probability plays an important role in almost all areas ofscience and technology. In modern mathematics, however, probability theorymeans nothing other than measure theory, and the operational characterizationof the notion of probability is not established yet. In this paper, based onthe toolkit of algorithmic randomness we present an operationalcharacterization of the notion of probability, called an ensemble. Algorithmicrandomness, also known as algorithmic information theory, is a field ofmathematics which enables us to consider the randomness of an individualinfinite sequence. We use the notion of Martin-Loef randomness with respect toBernoulli measure to present the operational characterization. As the firststep of the research of this line, in this paper we consider the case of finiteprobability space, i.e., the case where the sample space of the underlyingprobability space is finite, for simplicity. We give a natural operationalcharacterization of the notion of conditional probability in terms of ensemble,and give equivalent characterizations of the notion of independence between twoevents based on it. Furthermore, we give equivalent characterizations of thenotion of independence of an arbitrary number of events/random variables interms of ensembles. Moreover, we show that the independence of events/randomvariables is equivalent to the independence in the sense of van Lambalgen'sTheorem, in the case where the underlying finite probability space iscomputable. In the paper we make applications of our framework to informationtheory, cryptography, and the simulation of a biased coin using fair coins, inorder to demonstrate the wide applicability of our framework to the generalareas of science and technology.

An operational characterization of the notion of probability by algorithmic randomness and its applications