home email 

Leonhard Euler on Probability and Statistics

Leonhard Euler was born on 15 April 170 at Basel, Switzerland. He attended the university there where he made the acquaintance of Johann Bernoulli. He was invited to St. Petersburg in 1727  upon the death of Niklaus II Bernoulli where he worked with Daniel Bernoulli. Euler became professor of physics in 1731, and professor of mathematics in 1733, when he replaced Daniel who had departed for Basel that year. Euler was invited to Berlin by Frederick the Great of Prussia where he worked for the 25 year period from 1741. In 1766 Euler returned to St. Petersburg where he remained until his death on 18 September 1783.

Euler's works related to the calculus of probabilities and to statistics are all to be found in

Opera Omnia Series I, Volume 7, 1923
Edited. Louis Gustave du Pasquier

Available here are translations into English of the majority of these works together with references to various related documents. The papers are referenced by their Enestrom numbers. Of the few which have been previously translated, I cite the journals in which they appear. I thank A. Berra for the translation of E811 and Chapter VII of the De Usu of Nicolas Bernoulli; I am solely responsible for the remainder as well as for all errors.

These documents may be freely copied for use by educators and educational institutions as long as proper credit is given and they remain unaltered. This site may neither be mirrored nor its files reposted. Comments and corrections are welcome. As time permits I will expand the materials available here.

Euler's Notebook

Euler solves Huygen's Fifth Problem and investigates when the players have equal expectations in the Adversaria mathematica H4. This can only be dated approximately to the period 1740 to around 1750. 

The Game of Treize (Rencontre): E201

The game of Treize has sometimes been called Rencontre. Todhunter mentioned that the game or a generalization had been discussed by

Although Todhunter observed the later paper of Michaelis, he failed to note Catalan's paper, "D'un Problème de Probabilité, relatif au Jeu de rencontre," in Journal de Mathématiques Pures et Appliquées, 1837.

In the Essai d'analyse sur les jeux de hazard, 2nd edition (1713), Montmort discussed the game of Treize on pages 130 to 143. In the 1st edition of 1708 Montmort did not give a demonstration of any of his results. However, in the 2nd edition he gave two demonstrations which he had received from Nikolaus Bernoulli (pages 301 and 302). This includes a series with value e. Translation of Montmort.

In the 1st edition of the Doctrine of Chances (1717), the game is mentioned in the Preface and in Problems 35 & 36. These problems are contained in the 2nd edition of 1738 and the 3rd edition of 1756. The third edition contains the series for the constant e for the first time.

Euler's paper on the game of Rencontre, published in 1753, is E201, "Calcul de la probabilité dans le jeu de rencontre," Mémoires de l'académie de Berlin [7] (1751), 1753, p. 255-270. Regarding this work, the editor says that a memoir entitled "Calcul des probabilités dans les jeux de hasard" was presented to the Academy of Berlin 8 March 1758. He asserts that it is probably memoir 201: "Calcul de la probabilité dans le jeu de rencontre." An analysis of it appeared in the Nova Acta eruditorum, Leipzig 1754, p, 179.

Lambert produced a rather interesting paper entitled "Examen d'une espece de Superstition ramenée au calcul des probabilités" published in Nouveaux Mémoires de l'Académie Royale des Sciences et Belles Lettres for 1771. In this paper he cites Euler and gives a general solution for the number of encounters. However, the purpose of the paper is not to study this game but rather to give an argument to support of the "accuracy" of weather almanacs.

In Book II Chapter II Section 9 of the Théorie Analytic des  Probabilities, Laplace considers the following urn problem: In an urn are r balls labeled 1, r balls labeled 2, and so on to r balls labeled n. If the balls are drawn successively, what is the probability that at least one of the balls exits at the position indicated by its label? At least two? This same section treats a related problem: There are i urns each containing n balls, all of different colors. All the balls are drawn from each urn in turn. He asks, What is the probability that one or more balls will be drawn at the same rank from all the urns?

In Mémoire sur la probabilité du jeu de rencontre, G. Michaelis generalizes the two problems of Laplace in this way. He allows that the balls of each label vary in number.

Derangements: E738

Although not printed in volume 7 of his collected works, there is a paper on the number of derangements of a set of numbers which has a place here because of its relationship to E201. It is

E738. "Solution quaestionis curiosae ex doctrina combinationum" which was presented to the assembly 18 October 1779 and published in the Mémoires de l'académie des sciences de St. Pétersbourg 3 (1809/10), 1811, p. 57-64.

The Game of Pharaon: E313

The game of Pharaon was discussed by

The problem is to determine the advantage to the banker in the game. According to Todhunter, the problem was solved by Montmort and N. Bernoulli. However, the treatment of the problem by Montmort changed with edition. The second edition (1713) is more compact than the first (1708). In the second edition there is also a contribution by Jean III Bernoulli. Columns 2 and 4 of his tables are incorrect. Translation of Montmort.

The article from the Encyclopedia of Diderot on Pharaon may be compared to Montmort's treatment of the game. By the way, a punter is a gambler, the one who wagers against the banker.

Moivre asserts in his introduction that the chief question concerning Pharaon (and Bassette) is this: What percent does the banker receive of all the money wagered at these Games? He shows that for Pharaon it is nearly 3%, and  0.75%. for Bassette. Moivre on Pharaon.

In 1755, Euler wrote E313, "Sur l'avantage du banquier au jeu de Pharaon," Mémoires de l'académie de sciences de Berlin [20] (1764), 1753, p. 144-164. According to the editor, the memoir was presented 27 February 1755 to the Academy of Science of Berlin, and a second memoir bearing the same title was presented there 20 July 1758. It is not clear if these were the same work or two different works. The publication took place in 1766.

On lotteries: E338, E412, E600, E812, E813

The Genoise lottery was the first number lottery. It and its variants were discussed by many mathematicians because such lotteries were perceived to be unfair and because they gave rise to many interesting problems. Usually it took the form of choosing 5 from 100 with various payoffs depending upon the wager made. It was treated by

Juan Caramuel reprints the De ratiociniis in ludo aleae of Huygens in its totality. As for the Genoise lottery, he states that the payoffs are
Correct picks 1 2 3 4 6
Payoff 1 10 300 1500 10000

Caramuel argues that the corresponding correct payoffs are
8 10.7 334.1 31703.9 15090208

and hence the lottery is an unfair contract.

Nicolas Bernoulli, nephew of Jakob Bernoulli, submitted as his dissertation De Usu Arte conjectandi in jure, in partial fulfillment for the degree of Doctor of Laws. In Chapter VII he took up the subject of lotteries. There is strong evidence that Nicolas Bernoulli made use of a notebook in which Jakob himself considered the Genoise lottery.  He assumes the prizes should be inversely proportional to the probability of winning. In this way he obtains
0.95 10.9 337.3 31700 15057504

Because of his error, he treats Caramuel somewhat shabbily. Bernoulli's position is that this is an unfair contract as well.

Frenicle de Bessy died in 1675. The Paris Academy published his works in 1729. Among them is this short paper "Abregé des combinaisons." He analyzes the game for these payoffs
0 4 300 5000 20000

and claims the payoffs should be
0 0 500 or 600 5000 or 6000 20000


Euler's interest in lotteries began at the latest in 1749 when he was commissioned by Frederick the Great to render an opinion on a proposed lottery. The first of two letters began 15 September 1749. A second series began on 17 August 1763.

Euler himself wrote several papers prompted by investigations of lotteries.

E812. Read before the Academy of Berlin 10 March 1763 but only published posthumously in 1862. "Reflexions sur une espese singulier de loterie nommée loterie genoise." Opera postuma I, 1862, p. 319-335. The paper determined the probability that a particular number be drawn.

E338. "Sur la probabilité des sequences dans la loterie genoise." Mémoires de l'Académie royale des sciences et belles-lettres de Berlin [21] (1765), 1767, p. 191-230. As the name implies, Euler asks for the probability that various sequences of numbers be drawn.

The volume of the journal which contains E338, contains as well a paper by Jean III Bernoulli, "Sur les suites ou séquences dans la loterie de Genes," pp. 234-253 and two papers by Beguelin, "Sur les suites ou séquences dans la lotterie de Gene: First memoir and second memoir, " pp. 231-280.

E412. Read 29 November 1770. "Solution d'une questione tres difficile dans le calcul des probabilités." Mémoires de l'Académie royale des sciences et belles-lettres de Berlin [25] (1769), 1771, p. 255-302. This is an analysis of a lottery for which there are several classes and a guaranteed payment.

E600.  "Solutio quarundam quaestionum difficiliorum in calculo probabilis." Opuscula Analytica Vol. II, 1785, p. 331-346. Here Euler investigated the probability that all numbers or some fewer numbers be drawn in a sequence of lotteries.

Regarding this latter paper, see De Moivre, 1711, De Mensura Sortis Problem 18 or its nearly identical counterpart in the Doctrine of Chances Problem 39. In these places, de Moivre determined the expectation of one who would cast a die some number of times so as to produce all faces. P.S. Laplace  asked for the probability that all tickets will have been withdrawn after a prescribed number of drawings. This problem was solved in "Mémoire sur les suites récurro-récurrentes et sur leurs usages dans la théorie des hasards," Mém. Acad. R. Sci. Paris (Savants étrangers) 6, 1774, pages 353-371. Here Laplace refers to the Genoise Lottery as the Lottery of the Military School. Years later, in the Théorie analytique des Probabilités he asked for the number of drawings for which the probability that all tickets will have come forth is one-half. This is found in Book II, Chapter II, No. 4. The Genoise Lottery is now called the Lottery of France. Jean Trembley, citing the papers of both Euler and Laplace, generalized the solution to the problem in "Recherches sur une question relative au calcul des probabilités," Mémoires de l'Académie royale des sciences et belles-lettres, Berlin 1794/5, pp. 69-108.

E813 "Analyse d'un probleme du calcul des probabilites," Opera Postuma I, 1862, p. 336-341. In this paper, Euler determined the probability that a ticket will be drawn 0, 1, 2, ... times in n successive drawings of r tickets from an urn.

The St. Petersburg Problem: E811

The so-called St. Petersburg Problem began with a letter from Nicolas Bernoulli to Montmort. Eventually Daniel Bernoulli and Cramer became involved. The salient portions of there correspondence are here: Correspondence of Montmort, Daniel Bernoulli, Cramer and Nicolas Bernoulli.

Daniel Bernoulli is justly famous for his paper on risk in which he proposed a solution to the problem. This was published in 1738 as "Specimen theoriae novae de mensura sortis," Commentarii Acad. Sci. Imp. Petrop., 1730-31, 5, 175-192. It has been translated into English as "Exposition of a new theory on the measurement of risk" Econometrica, 1954, 22, 23-36.

Euler wrote a short paper quite similar in content to that of Daniel Bernoulli. Its time of composition is unknown.

E811. "Vera estimatio sortis in ludis." Opera postuma I, 1862, p. 315-318.

More documents related to this problem are located in pages devoted to D'Alembert.

Errors of Observation: E448, E628

Many mathematicians were concerned with the problem of how to combine discordant observations. One common method was to compute the arithmetic mean. An early paper on this subject was that written by Thomas Simpson. In the work Miscellaneous Tracts on some curious, and very interesting Subjects in Mechanics, Physical-Astronomy, and Speculative Mathematics... is a section An Attempt to shew the Advantage arising by Taking the Mean of a Number of Observations, in Practical Astronomy. This item may be found at the University of York. But other methods could be justified. Daniel Bernoulli wrote on this problem in "Diiudicatio maxime probabilis plurium observationem discrepantium atque verisimillima inductio inde formanda." Acta Acad. Sci. Imp. Petrop., 1777 (1778), 1, 3-23. It has been translated into English by C.G. Allen as "The most probable choice between several discrepant observations and the formation therefrom of the most likely induction," Biometrika, 1961, 48, 1-18.

Euler wrote a commentary to Bernoulli's paper which was published in the same issue of the journal. It is

E448. (1778) "Observationes in praecedentem dissertationem illustris Bernoulli." Acta. Acad. Sci. Imp. Petrop. 1, 24-33 (1777). This likewise has been translated into English by C.C. Allen as "Observations on the foregoing dissertation of Bernoulli" Biometrika 48, 13-18, (1961).

Joseph Louis Lagrange wrote  "Mémoire sur l'utilité de la méthode de prendre milieu entre les résultats de plusiers observations," Miscellanea Taurinensia, t. V, 1770-1773. Euler responded to this paper with

E628. "Eclaircissements sur le mémoire de Mr. De la Grange insere dans le Ve volume des melanges de turior concernant la methode de prendre le milieu entre les resultats de plusiere observations etc.," Nova acta academiae scientiarum Petropolitanae 3 (1785), 1788, p. 289-297. Summary p. 196-197. The paper was presented to the Academy 27 Nov. 1777.

Mathematical Statistics: E334

Euler concerned himself with mortality and life expectancy in E334. "Recherches générales sur la mortalité et la multiplication du genre humain." Mémoires de l'Académie royale des sciences et belles-lettres de Berlin [16] (1760), 1797, p.144-164. But see also the section on Life Assurance for the companion piece E335, "Sur les rentes viageres." and the mortality table of Kersseboom.

There are two related items included in his collected works. These are

"Von der geschwindigkeit der vermehrung und von der Zeit der verdoppelung." which was printed in Süssmilch's Die göttliche Ordnung and "On the multiplication of the human race." being a fragment from one of his notebooks.

Life Assurance: E335

Immediately following the memoir on mortality (E334) is the memoir

E335. "Sur les rentes viageres," Mémoires de l'Académie royale des sciences et belles-lettres de Berlin[16] (1760), 1797, p.165-175. Euler viewed this paper as a continuation of E334. Here he derived a formula to facilitate the computation of a life annuity. Euler observed that there is no advantage to the state to sell annuities where the return is greater than the rate of interest earned by the state. Therefore he proposed the creation of foreborne annuities, purchased, for example, on a child at birth, but due at age 20. This would then permit the accumulation of funds as well as allow the annuity to be offered at a much lower rate.

The collected works include as well

E403 Des Herrn Leonard Eulers nöthig berechnung zur Einrichtung Einer

E473 D'un Etablisement publie pour payer des pensions a des veuves fonde sur les principles les plus solides de la probabilité. pp. 183-245. This memoir includes a description of a tontine.

E599 Solutio quaestiones ad calculam probabilitatis pertinentis quantiam disconceges persolvere debeant ut suis haeradibus post utriusque mortem certa argenti summa persolvatur. Opuscula Analytica II, 1785, p. 315-330.

Fragment. Sur le calcul des rentes Tontinieres.

This page was last updated on 08/20/11.