# Evolution of teaching the probability theory based on textbook by V. P. Ermakov

### Abstract

*The paper is devoted to the study of what changes the course of the probability theory has undergone from the end of the 19 ^{th }century to our time based on the analysis of The Theory of Probabilities textbook by Vasyl P. Ermakov published in 1878. In order to show the competence of the author of this textbook, his biography and creative development of V. P. Ermakov, a famous mathematician, Corresponding Member of the St.*

*Petersburg Academy of Sciences, have been briefly reviewed. He worked at the Department of Pure Mathematics at Kyiv University, where he received the title of Honored Professor, headed the Department of Higher Mathematics at the Kyiv Polytechnic Institute, published the Journal of Elementary Mathematics, and he was one of the founders of the Kyiv Physics and Mathematics Society. The paper contains a comparative analysis of The Probability Theory textbook and modern educational literature. V. P. Ermakov's textbook uses only the classical definition of probability. It does not contain such concepts as a random variable, distribution function, however, it uses mathematical expectation. V. P. Ermakov insists on excluding the concept of moral expectation accepted in the science of that time from the probability theory. The textbook consists of a preface, five chapters, a synopsis containing the statements of the main results, and a collection of tasks with solutions and instructions. The first chapter deals with combinatorics, the presentation of which does not differ much from its modern one. The second chapter introduces the concepts of event and probability. Although operations on events have been not considered at all; the probabilities of intersecting and combining events have been discussed. However, the above rule for calculating the probability of combining events is generally incorrect for compatible events. The third chapter is devoted to events during repeated tests, mathematical expectation and contains Bernoulli's theorem, from which the law of large numbers follows. The next chapter discusses conditional probabilities, the simplest version of the conditional mathematical expectation, the total probability formula and the Bayesian formula (in modern terminology). The last chapter is devoted to the Jordan method and its applications. This method is not found in modern educational literature. From the above, we can conclude that the probability theory has made significant progress since the end of the 19*

^{th}century. Basic concepts are formulated more rigorously; research methods have developed significantly; new sections have appeared.### Downloads

### References

Bilová, Š., Mazliak, L., & Šišma, P. (2006). The Axiomatic melting pot: Teaching probability theory in Prague during the 1930’s. Journal Electronique d’Histoire des Probabilités et de la Statistique, 2(2), 30. Retrieved from https://arxiv.org/pdf/math/0607217.pdf [in French].

Dobrovolsky, V. A. (1981). Vasilii Petrovich Ermakov [Vasily Petrovich Ermakov]. Moscow: Nauka [in Russian].

Fischer, H. (1994). Dirichlet's contributions to mathematical probability theory. Historia mathematica, 21, 39–63.

Flood, R., Rice, A., & Wilson, R. (2011). Mathematics in Victorian Britain. Oxford: Oxford University Press.

Gnedenko, B. V. (1988). Kurs teorii veroiatnostei [Probability Theory Course]. (6th ed., rev.). Moscow: Nauka. [in Russian].

Gnedenko, B. V., & Gikhman, I. I. (1956). Razvitie teorii veroyatnostej na Ukraine [Development of Probability Theory in Ukraine]. Istoriko-matematicheskie issledovaniia – Historical-mathematical research, 9, 477–536 [in Russian].

Gmurman, V. E. (2004). Teoriia veroiatnostei i matematicheskaia statistika [Probability Theory and Mathematical Statistics]. 10th ed. Moscow: Vysshaia shkola [in Russian].

Gorban, S. F., & Snizhko, N. V. (1999). Teoriia veroiatnostei i matematicheskaya statistika [Probability Theory and Mathematical Statistics]. Kyiv: MAUP [in Russian].

Gratsianskaia, L. N. (1956). Vasilii Petrovich Ermakov [Vasily Petrovich Ermakov]. Istoriko-matematicheskie issledovaniia – Historical-mathematical research, 9, 667–690 [in Russian].

Kartashov, M. V. (2008). Imovirnist, protsesy, statistika [Probability, Processes, Statistics]. Kyiv: Vidavnicho-poligrafichnii tsentr «Kyjivskyi universytet» [in Ukrainian].

Klesov, O. I. (2010). Vybrani pytannia teorii ymovirnostei ta matematychnoi statystyky [Selected Issues of the Probability Theory and Mathematical Statistics]. Kyiv: TViMS [in Ukrainian].

Kletska, T. S. (2016). Istorychni vytoky Kyivskoi shkoly teorii ymovirnostei [The Historical Origins of the Kyiv School of Probability Theory]. Istoriia nauky i tekhniky – History of science and technology, 8, 123–129 [in Ukrainian].

Kolmogorov, A. N., & Yushkevich, A. P. (Eds.). (1978). Matematika XIX veka. Matematicheskaya logika. Algebra. Teoriya chisel. Teoriya veroyatnostej [Mathematics of the 19th century. Mathematical Logic. Algebra. Number Theory. Probability Theory]. Moscow: Nauka [in Russian].

Kushlyk-Dyvulska, O. I., Polishchuk, N. V., Orel, B. P., & Shtabaliuk, P. I. (2014). Teoriia ymovirnostei ta matematychna statystyka [Probability Theory and Mathematical Statistics]. Kyiv: NTUU “KPI” [in Ukrainian].

Mačák, K. (2005). Vývoj teorie pravděpodobnosti v českých zemích do roku 1938 (krátký přehled). Praha: Prometheus. Retrieved from https://dml.cz/handle/10338.dmlcz/401182 [in Czech].

Meusnier, N. (2006). Sur l’histoire de l’enseignement des probabilités et des statistiques. Journal Electronique d’Histoire des Probabilités et de la Statistique, 2(2), 20. Retrieved from https://www.emis.de/journals/JEHPS/Decembre2006/ Meusnier.pdf [in French].

Psheborskyi, A. P. (1922). V. P. Ermakov [V. P. Ermakov]. Nauka na Ukraine – Science in Ukraine, 3, 285–287 [in Russian].

Rabyk, V. M. (2004). Osnovy teorii ymovirnostei [Foundations of the Probability Theory]. Lviv: Mahnoliia plius [in Ukrainian].

Seno, P. S. (2007). Teoriia ymovirnostei ta matematychna statystyka [Probability Theory and Mathematical Statistics]. Kyiv: Znannia [in Ukrainian].

Shiryaev, A. N. (1979) Veroiatnost [Probability]. Moscow: Nauka [in Russian].

Sliusarchuk, P. V. (2005). Teoriia ymovirnostei ta matematychna statystyka [Probability Theory and Mathematical Statistics]. Uzhhorod: Vydavnytstvo “Karpaty” [in Ukrainian].

Ermakov, V. P. (1879). Teoriia veroiatnostei: lektsii chitannyia v Imperatorskom Universitete Sv. Vladimira professorom V. P. Ermakovym. Ottisk iz Universitetskikh izvwstiy 1878 g. [The Probability Theory: Lectures Given at the Imperial University of St. Vladimir by prof. V. P. Ermakov. An Imprint from the University News of 1878]. Kyiv: Tipographiya Imoeratorskago Universiteta Sv. Vladimira [in Russian].

Yezhov, S. M. (2001). Teoriia ymovirnostei, matematychna statystyka i vypadkovi protsesy [Probability Theory, Mathematical Statistics, and Random Processes]. Kyiv: VPTs “Kyivskyi universytet” [in Ukrainian].

*History of Science and Technology*,

*11*(2), 300-314. https://doi.org/10.32703/2415-7422-2021-11-2-300-314

**License terms:** authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a **Creative Commons Attribution License International CC-BY **that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.

The scanned copy of the **"Agreement” on the authors" **copyright transfer on the manuscript publication and the subsequent posting of the paper on the Internet (in * .pdf or * .jpg format) is to be attached to the manuscript of the paper.

**By this agreement the author certifies that the submitted material:**

- does not infringe the copyright of other persons or organizations;
- was not previously published in other publishing houses and has not been submitted for publication in other editions.

**The author passes the editorial board of the journal "History of science and technology" rights to:**

- publication of the article in Ukrainian (English and Russian) language and distribution of its printed copy;
- translation of the article into English language (for articles in Ukrainian and Russian language) and distribution of its translated printed copy;
- distribution of the article electronic copy, as well as electronic copy of the article English translation (for articles in Ukrainian and Russian), via any electronic means (placing on the official web-site of the journal, electronic databases, repositories, etc.) printed copy of the translation.

**The author reserves the right without the consent of the editorial board and founders:**

- Use the materials of the article in whole or in part for educational purposes.
- Use the materials of the article in whole or in part to write their own dissertations.
- Use the materials of the article for the preparation of abstracts, conference reports, as well as oral presentations.
- Place electronic copies of the article (including the final electronic copy downloaded from the official web-site of the journal) to:

- personal web-resources of all authors (web-sites, web-pages, blogs, etc.);
- web-resources of institutions where authors work (including electronic institutional repositories);
- non-commercial web-resources of open access (for example, arXiv.org).

In all cases, the availability of a bibliographic link to an article or hyperlink to its electronic copy on the official website of the journal is compulsory.