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A Selection of Problems from A.A. Markov’s Calculus of Probabilities: Markov's Book

Alan Levine (Franklin and Marshall College)


Title page of the 1900 first edition of Calculus of Probabilities.
Figure 3. Title page of the first edition. GoogleBooks.

As noted in the overview, the four editions of Markov's Calculus of Probabilities appeared in 1900, 1908, 1912, and, posthumously, 1924. Parts of the second and third editions have been translated into French and German, but there appears to have been no English translation of any of the editions until now. The first edition (from 1900) contains eight chapters.3

Table of Contents for Markov’s Calculus of Probabilities (1st ed., 1900)

Chapter I Basic concepts and theorems 1–20
Chapter II On repeated experiments 21–51
Chapter III On the sum of independent variables 52–100
Chapter IV Examples of various methods of calculating probabilities 101–157
Chapter V Limits, irrational numbers and continuous variables in the calculus of probabilities 158–187
Chapter VI Probability of hypotheses and future events 188–212
Chapter VII The method of least squares 213–266
Chapter VIII On life insurance 267–279

The fourth edition (of 1924) is approximately twice the length of the first edition. Much of the extra material is in Chapter VII, which tripled in size. There is some additional material—which he could have called “chapters”—on limit theorems, method of moments, and experiments connected in a chain. Since this edition was published after Markov's death, it contains a biographical sketch of Markov written by his student, Abram Bezicovich (1891–1970). I have included that sketch in my full translation of the first edition (pp. 9–17).


[3] A translation of the entire first edition can be found at:

About the translation, it is fair to say that Russian does not translate smoothly into English. Russian word order is quite flexible and punctuation rules are different. The lack of the present tense of the verb “to be” leads to excessive use of passive voice. I have tried to translate the work as literally as possible, within the bounds of sensible English.

The original Russian first edition of the text can be found at this link. The Russian language was simplified a bit after the Russian Revolution in 1917. Some letters were removed from the alphabet and there were some spelling changes. Most notably, the “hard sign”, ъ, (твёрдый знак), which appeared at the end of many words that ended in a consonant, was eliminated (although it can still be found in the middle of some words). The fourth edition incorporates many of these modifications. I have retyped the first edition in the modern Russian language. It is available upon request.


Alan Levine (Franklin and Marshall College), "A Selection of Problems from A.A. Markov’s Calculus of Probabilities: Markov's Book," Convergence (November 2023)