Euler's Analysis of the Genoese Lottery - Probabilities in the Genoese Lottery

That the Genoese lottery captured Euler's mathematical imagination is evidenced by the entries he made in his notebook, known today as H5, which he probably filled between 1748 and 1750 [6, p. xxiv]. His first published article on the lottery did not appear until 1767. However, the Genoese lottery was only established in Berlin in 1763, and on March 10 of the same year, Euler delivered an address entitled "Reflections on a Singular Type of Lottery called the Genoese Lottery" [11] to the Berlin Academy. The text of the address was finally published in 1862, in Euler's Opera Posthuma I. Leonard Euler's publications were numbered from E1 through E856, in the early 20th century by Gustav Eneström, in the order of their publication. The fact that this paper was published posthumously is reflected in its high number in Eneström's index: E812.

Unlike Euler's later papers on the Genoese lottery, which are works of pure mathematics that set out to answer questions of little or no practical value inspired by the lottery, E812 is a work of applied mathematics. Euler proves little new mathematics, but instead applies elements of probability theory, interspersed with dashes of common sense and vaguely justified rules of thumb, in describing how one might go about designing a Genoese style lottery from scratch, determining, in particular, fair prize levels. This is a different undertaking from his analysis of Roccolini's proposal, where the prize amounts were given, and Euler determined the fair price of a ticket, comparing these to Roccolini's prices.

Euler's goal in the first portion of the paper is to calculate the probability p_k,i that a player who bets on k numbers will in fact match i of them (Euler did not use the notation p_k,i; we are adopting it here to make the discussion easier to follow). The values depend on the parameters n and t, where the lottery consists of choosing t tokens at random from a collection numbered 1, 2, 3, ¼, n. A modern probability text would say that i has hypergeometric distribution, with parameters n, t and k, and thus

pk,i = æ ç ç è t i ö ÷ ÷ ø æ ç ç è n-t k-i ö ÷ ÷ ø æ ç ç è n k ö ÷ ÷ ø .

However, the hypergeometric distribution, the distribution of sampling without replacement, did not yet have the status of a standard random variable in Euler's time. Euler's analysis was indeed one of the first treatments of this distibution in print, although Jacob Bernoulli and Abraham de Moivre had considered the hypergeometric distribution earlier in the 18th century [12, pp. 201-202]. Like Bernoulli and de Moivre, Euler did not give the p_k,i's in the above form; in fact, he had not yet even developed a convenient notation for binomial coefficients, although he would develop a notation similar to the modern one later in his career. Instead, Euler gave the p_k,i in a form that was better suited to efficient recursive calculation.

Euler gives a complete derivation of the desired probabilities. His method of proof - an implicit or `socratic' induction - is a common one in Euler's writings: he solves the simplest cases in order, clearly and persuasively argued, until the pattern is clear to the reader. Paper E812 is divided into sections which Euler calls Problems, some with corollaries or scholia. He considers the distribution of i in the case k=1 in Problem 1, and then proceeds through the next three values of k in Problems 2-4. We summarize these in Table 2, where the kth column represents the results of the corresponding Problem.

pk,i	k = 1	k = 2	k = 3	k = 4
i=0	n-t n	(n-t)(n-t-1) n(n-1)	(n-t)(n-t-1)(n-t-2) n(n-1)(n-2)	(n-t)(n-t-1)(n-t-2)(n-t-3) n(n-1)(n-2)(n-3)
i=1	t n	2t(n-t) n(n-1)	3t(n-t)(n-t-1) n(n-1)(n-2)	4t(n-t)(n-t-1)(n-t-2) n(n-1)(n-2)(n-3)
i=2		t(t-1) n(n-1)	3t(t-1)(n-t) n(n-1)(n-2)	6t(t-1)(n-t)(n-t-1) n(n-1)(n-2)(n-3)
i=3			t(t-1)(t-2) n(n-1)(n-2)	4t(t-1)(t-2)(n-t) n(n-1)(n-2)(n-3)
i=4				t(t-1)(t-2)(t-3) n(n-1)(n-2)(n-3)
Table 2: Summary of Problems 1-4