Page 101 - DP-2-2

Design+ Modern interpretations of probability

At the same time, in the general case, when the exact
value of probability based on the property of symmetry
of events cannot be calculated, an inductive assumption is
made that the relative frequency of such mass, recurrent
events is very close to some number, which is called
probability. Consequently, inductive inference is necessary
to support the frequency interpretation of probabilities.
Still, given that any inductive inference is a departure
from the boundaries of the known, it inevitably leads to
difficulties of a theoretical nature.
An essential (almost the most important) problem
arising in applying methods based on the frequency
interpretation of probability is the verification of the
correctness of the obtained results. This is because
verifying the truth (adequacy) of the obtained probabilistic
Figure 3. Richard Edler von Mises. Image obtained from the Wikidata conclusion is quite natural in the study of any events
website. 7
occurring both in animate and inanimate nature. In
general, neither the verification nor the falsification
out. This artifact was based on the relative frequency of procedure can verify this.
8
the realization of a mass random event with sufficiently
long observations. At the same time, the probability value If we consider classes of events as sets of finite power,
in all such situations was determined based on statistical then probabilistic inference would directly indicate the
procedures and was called statistical probability. Given that arithmetic ratio of the powers of these classes. In this case,
statistical probability is numerically determined by relative the probabilistic inference could be checked by presenting
frequency, it is often called frequency probability. This all elements of both classes, followed by a simple count. If
approach is adopted in statistics today, where probability is we consider infinite classes of events or an infinite number
identified with the relative frequency of the realization of a of finite statistical sequences, then no finite number of
mass random event with sufficiently long observations. Note observations can definitively confirm or definitively
that in the definition of this interpretation of probability reject the probabilistic conclusion made. One cannot (at
by Mises (and Venn), the duration of observations is not least theoretically) exclude that a given finite sequence of
predetermined. Instead, it is proposed to be established observations is a certain fluctuation, a significant deviation
during the research process, that is, simultaneously with of the relative frequency in a given sequence from the
the observation of the analyzed event and based on the relative frequency in the entire infinite class of analyzed
data collected through these observations. In addition, the events.
conditions for realizing events must remain unchanged In this context, it should be noted that the common
to obtain an adequate result throughout the observation practice among specialists from different fields of discarding
process. individual observations (experiments) or experimental

Statistical probability is the limit value of the sequence results that differ significantly from the results of the bulk
of relative frequencies of the realization of the expected of observations realized within the framework of this
event among all observed and analyzed events. However, study, regardless of circumstances, cannot be recognized
this limit remains theoretically unproven and does not as methodologically flawless. Such an exclusion procedure
allow us to accept Mises’s constructions as an appropriate should be treated with caution, applying it only after a
theory. thorough analysis of the relevant situation and establishing
the reasons, if possible, for this difference.
Long-term practice has established that for many mass
phenomena (for streams of events), the relative frequency An important problem related to frequency probability
tends to stabilize as the number of observations increases. is the logic of constructing observations. This problem
Such stability of frequencies of mass phenomena is an highlights the necessity of a clear formulation of the
objective regularity, independent of human will and desire. conditions that sequences of events must satisfy for the
The practice also shows that if it is possible to calculate the frequency interpretation of probability to be validly applied.
probability of events based on the property of symmetry This is because the calculated frequency-based probability
of these events, then the relative frequency is quite an often depends significantly on the sequence of realization
acceptable estimate of the specified probability. of events of a particular population (on the organizational

Volume 2 Issue 2 (2025) 6 doi: 10.36922/dp.6387