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Design+ Modern interpretations of probability

definitions of probability and its meanings. Probability is a
number P (A) ϵ [0,1] characterizing the degree of possibility
of occurrence of the determining event A. For example,
P (X = a) means the probability that the random variable X
takes the value a.

Classical probability is the ratio of the number of
cases favorable to a given event to the total number of
equally likely cases. The prior probability of event B і
is the probability of event P(B ) before the experiment.
і
Conditional probability is the probability of event A, given
that some other event Bі, P (A/B), is carried out at the same
time. The posterior probability is the probability of event B і
after an experiment in which event A occurred, P (B /A).
і
Bayes’ theorem relates the categories of prior conditional
Figure 4. Andrej Nikolayevich Kolmogorov. Image taken from and posterior probabilities: if B , B .,B is a complete system
n
1
2
Wikipedia. 15 of events, P (B ) is the prior probability of events B , and
k
k
P (A/B ) is the conditional probability of event A given
k
event A on the set of all events determined by a series that event B has occurred, then the posterior probability
k
of corresponding observations. It is assumed that the P (B /A) of event B given that event A has occurred is
k
k
specified set contains not only all events but also all possible defined in Equation I.
combinations of these events, and the introduced function
( ) PA⋅
satisfies the conditions 0 ≤ P (A) ≤ 1, and P (A) = 1 if A is a P B k ( / )B k
valid event. Besides, the truth of the relation P (A v B) = P ( PB k / A ) = ∑ n PB (I)
( ) ( / A B⋅
(A) + P (B) is accepted, which makes the simultaneous j= 1 j j
realization of events A and B impossible. What is important
in the above axiomatics is that it is considered on the set of This expression reflects the so-called Bayesian approach
events whose measure equals one. to statistical problems.
This interpretation of probability describes the formal The difference between the Bayesian approach and
properties of this concept, built on mathematical artifacts, other statistical approaches is that even before the data
operates with such artifacts, and, based on them, leads to are obtained, the researcher considers their degree of
certain conclusions. Such a theory says nothing about the confidence in possible models and represents it in the form
nature of the phenomena to which it is applied and which of prior probabilities.
it describes. In this interpretation, the probability is a Geometric probability is the probability of hitting a
function taking values in the interval from zero to one, and certain part of the area on which a uniform distribution is
satisfying appropriate conditions, as the events themselves defined. The statistical probability is the relative frequency
must fulfill appropriate conditions. All these conditions and of events in a sufficiently long series of experiments.
properties are introduced to achieve the appropriate logical Subjective probability is the probability of an event
structure of the theory, and its internal incontestability, according to the subjective opinion of an expert. In the
aimed at “servicing” the theory. In practice, the events and axiomatic interpretation of probability, the key is the
the specified function may not correspond (and often do definition of the concept of random events, which are
not correspond) to the established conditions. Notably, formalized using a probability space that is given by the
the abstract concept of probabilities represents certain triad (Ω, Ψ, and P), where Ω is the space of elementary
properties of frequencies observed within the framework events ω ϵ Ω, Ψ is the σ-algebra of subsets of events, which
of the corresponding observations, but they are not is defined as the algebra of sets S containing the union of
completely reduced to these frequencies. At the same time, any number of its elements (or Borel field), and P is the
the mathematical apparatus of probability theory (in its probability measure (probability) of a subset of events.
axiomatic interpretation) is widely and fruitfully used in The accurate use of probability theory is provided
the practical activities of specialists in various technical by accepting the physical hypothesis of ideal statistical
and humanitarian knowledge fields.
stability (stability) of parameters and characteristics of
The widely developed modern mathematical physical phenomena. However, experimental studies
interpretation of axiomatic probability provides standardized on large observation intervals of various processes of

Volume 2 Issue 2 (2025) 8 doi: 10.36922/dp.6387