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Design+ Modern interpretations of probability

basis of observations). This leads to the philosophical hyper-random events, providing a strong rationale for
conclusion that to obtain an adequate result, the frequency viewing the development of this branch of probability
interpretation of probability should be applied not to the theory, particularly within its frequency interpretation, as
fact of realization of events of the studied population, but a promising direction.
rather to the organization of the process of realization of
these events, that is, to the process of research and the 5. Axiomatic interpretation of probability
situation “accompanying” this research. The axiomatic approach to constructing probability theory
Here, recalling the dispositional interpretation of is a turning point in developing this branch of mathematical
probability proposed by the Austrian–British philosopher science. The introduction of axiomatics, as the basis of
and sociologist Karl Raimund Popper is appropriate. all further conclusions and provisions, turns probability
According to this interpretation, probability is not a property calculus into a rigorous mathematical discipline, free from
of events in the investigated population but determines the implicit assumptions and logical contradictions. However,
dispositional properties of some (preliminarily accepted) the flawlessness of the theory was achieved at the expense
investigated situation. of a complete abandonment of any substantive laziness
Unlike the statistical one, the dispositional interpretation concerning the concept of probability itself. Within the
of probability gives probability an ontological status axiomatic theory, probability as a concept is not defined or,
because probability is an inherent property of this or that more precisely, is defined formally as some appropriately
objective situation (disposition), which can manifest itself specified function, a mathematical artifact satisfying an
in various forms regardless of our knowledge. Popper’s established system of axioms. In this case, the question is
conclusions have not found wide support in the scientific not even posed as “what is the probability of a real event?”,
community (mainly about the possibility of applying the and how to calculate or measure this probability in the
concept of frequency probability to assess the likelihood sphere of practical activity is not considered.
of the realization of single events). However, they should The axiomatic interpretation (in this case, the word
be considered when preparing studies, the results of which “theory” is also quite applicable) of probabilities is related
are supposed to be processed based on the concept of to an idealized mathematical reality. Among the existing
frequency probability. axiomatic theories of probabilities, the most general and
Thus, the frequency probability should be considered transparent one is considered to be the theory proposed by
as a mathematical artifact characterizing quantitative Andrej Nikolayevich Kolmogorov in 1929. Kolmogorov’s
regularities (ratio) of two sets of events and cannot be applied axiomatic probability theory 13,14 is considered in
when we are talking about non-repeating, rarely repeated, mathematics as a separate direction of measure theory,
or single events. In practice, it is usually impossible to fulfill relying on the concepts and methods of set theory. Within
the conditions that would give confidence in the existence the framework of this axiomatic theory, the very concept of
of a limit value of the relative frequency of the realization probability is considered an undefined initial concept. The
of events corresponding to the frequency probability. In motives for constructing an axiomatic theory of probability
particular, in observing events, it is practically impossible and his attitude to such a theory with A.N. Kolmogorov are
to ensure the consistency of the conditions in which these presented in the following words:
observations are carried out. This always gives certain reasons Probability theory as a mathematical discipline can
to doubt that the obtained value of the relative frequency and should be axiomatized completely in the same
of realization of events, even with its relative stabilization sense as geometry and algebra. This means that once
within the framework of the observations, will adequately the names of the objects to be analyzed and their basic
reflect the probability of realization of the analyzed event relations have been given, and the axioms to which
when the number of observations rises to infinity.
these relations must satisfy have been introduced, all
The existence and practical necessity of studying further teaching of probability theory must be based
events that can be observed exclusively in constantly solely on the axioms introduced, without relying on
changing conditions lead to the concept of hyper-random the ordinary concrete meaning of the objects and
events. 9-12 A characteristic feature of such events is the their relations indicated. 13(p.1)
impossibility of applying the described idea of frequency The portrait of Andrej Nikolayevich Kolmogorov is
probability because any increase in the observed number shown in Figure 4. 15
of event realizations will not limit the frequency of event
realization under the specified conditions. At the same In the framework of Kolmogorov axiomatics, the
time, numerous applied problems involve the study of probability is defined as a numerical function P(A) of an

Volume 2 Issue 2 (2025) 7 doi: 10.36922/dp.6387