You are here

Euler's Analysis of the Genoese Lottery

Robert E. Bradley
His industry and genius have left a permanent impression in every field of mathematics; and although his contributions to the Theory of Probability relate to subjects of comparatively small importance, yet they will be found not unworthy of his own great powers and fame.
Isaac Todhunter on Leonard Euler [1]

Biographies of Leonhard Euler (1707-1783) are widely available; Dunham [3, pp. xix-xxviii] and Calinger [2, pp. 486-489] have excellent lively accounts, while the Dictionary of Scientific Biography [4, pp. IV.467-484]DSB has a longer and more scholarly entry. Euler published over 30,000 pages of mathematics and physics, currently available in the 74 volumes of the first three series' of his Opera Omnia[5]. Volume 7 in Series I of the Opera Omnia is probably one of the most often consulted volumes of the entire series, because it contains papers on a variety of well-known mathematical problems, including the Bridges of Königsberg, the Knight's Tour, Magic Squares, the problem of Derangements and the Josephus Problem. However, these gems of recreational mathematics are to be found nearly buried among the more prevalent subject matter of this volume: some two dozen entries concerning probability theory and related subjects.

"Towards the middle of his life," wrote Louis Gustave du Pasquier, the editor of volume I.7, "Euler devoted a portion of his universal interest to the study of the theory of risk and ¼ to questions involving the calculus of probability [6, xxiii]." Alongside articles on observational error, mathematical statistics and the foundations of life insurance, volume I.7 contains eight memoirs and a fragment concerning probability theory on finite sample spaces. All of these are inspired by games of chance, be it the casino game Pharaon, the card game Rencontre, or the well-known Petersburg Problem. However, the greatest portion of Euler's writings on probability theory relate to the Genoese lottery.

Lotteries, the drawing of prizes "by lot," are as old as the written word. A very popular form of lottery today involves placing numbered balls in a large wheel, and drawing five or six at random. Players who correctly guess all or most of the numbers so drawn win cash prizes that are often worth millions of dollars. The name "Genoese Lottery" was commonly used in the 18th century for such a game of chance, because the game originated in the Italian city of Genoa in the 17th century, where participants would bet on the drawing of five balls from a ruota, or wheel, containing balls numbered 1, 2, 3, ¼, 90. The game originated from the election of city councillors by lot. Du Pasquier describes the origins of this game in [6, xxiv], but modern scholarship calls some of the details of this into question. A more recent account of the origins of the Genoese Lottery can be found in the article by Bellhouse [7].