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.
Biographies of Leonhard Euler (17071783) are widely available; Dunham [3, pp. xixxxviii] and Calinger [2, pp. 486489] have excellent lively accounts, while the Dictionary of Scientific Biography [4, pp. IV.467484]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 wellknown 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 wellknown 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].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.
Biographies of Leonhard Euler (17071783) are widely available; Dunham [3, pp. xixxxviii] and Calinger [2, pp. 486489] have excellent lively accounts, while the Dictionary of Scientific Biography (DSB) [4, pp. IV.467484] 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 wellknown 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 wellknown 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 councilors by lot. Du Pasquier describes the origins of this game in [6, xxiv], but modern scholarship calls some of his details into question. A more recent account of the origins of the Genoese Lottery can be found in the article by Bellhouse [7].
In the middle of the 18th century, the Italian craze for lotteries swept the European continent. It was thus that an Italian named Roccolini approached Frederick the Great of Prussia in 1749 with a scheme to establish a Genoesestyle lottery in Berlin. This was some eight years after Leonard Euler had come to Frederick's court from St. Petersburg. The king, as was his custom when mathematical matters were involved, called upon Euler for counsel.
Euler was already working on a royal assignment when Frederick's letter of 15 September 1749 arrived, along with a copy of Roccolini's proposal for the Berlin lottery. Euler set aside the earlier assignment  the design of a new hydraulic system for Frederick's summer palace, SansSouci  and two days later returned his analysis of the proposal to the king.
Roccolini's proposal involved the usual drawing of 5 numbers from a list of 90 and offered three principal ways for gamblers to place their wagers:It was also possible to place mixed bets, but apparently the proposal didn't specify the price of such tickets. Frederick's charge and Euler's response are reproduced in Volume IVA.6 of the Opera Omnia [10, pp. 316320], but unfortunately Roccolini's proposal is not included.
Euler's analysis began with a calculation of the fair price of each type of ticket, "according to the law of equality". In other words, he calculated the expected value of the payoff in each case. Table 1 summarizes his findings. In this 1749 letter, Euler determined the bank's profit using the formula


Type  Probability  Expected  Price of  The Bank's  The Bank's  
of bet  of a win  Prize  Value  a Ticket  Profit (1749)  Profit (1763)  
simple 

100  5 5/9  8  44%  31%  
ambo 

120  0.2996  0.5833  95%  49%  
terno 

180  0.01532  0.05208  240%  71%  
Table 1: Summary of Euler's Analysis 
Euler's analysis continued with a discussion of the fairness of the bank's increasing profit margin. It is important to note here a significant difference between the Genoese lottery and most modern versions of the lottery. The Genoese lottery (and, for that matter the contemporary Italian Lotto) offers fixedodds payoffs. That is, the player knows in advance how large a prize is at stake, irrespective of the number or distribution of tickets sold. In the long run, the bank is guaranteed to win, and to win big. However, if the state were to have a run of bad luck, and a relatively large number of major prizes were to be awarded in a drawing where relatively few tickets had been sold, the bank could, in principle, be broken.
The danger of bankruptcy is avoided in lottery games played today in the USA and most other countries by adopting a version of the method of parimutuel betting used in horse racing. The term, literally meaning `mutual wager', comes from two French words. A portion of each wager placed is set aside to cover the cost of running the lottery and the state's revenue, and the remainder is placed in a pool, to be paid out to the winners. In effect, players are betting against each other, with the state taking a cut of every wager.
In New York State's Lotto, for example, 6 winning numbers are chosen from a wheel containing 59 numbered balls. Players choose 6 numbers between 1 and 59, and win a cash prize if 3, 4, 5 or 6 of their chosen numbers match the winning numbers drawn from the wheel. (As an added wrinkle, a seventh "bonus number" is also drawn from the wheel, and a special second prize goes to any player who matches 5 of the 6 regular numbers drawn, plus the bonus number.) Thirtyeight cents out of every onedollar ticket goes into a parimutuel pool, and it is from this pool that all prizes are paid. Threequarters of that amount is reserved for the jackpot, paid to players who match all 6 numbers of the winning numbers. If there is no jackpot winner in a particular drawing, the jackpot rolls over, making the subsequent drawing even more attractive to players. Reliable estimates on the size of the jackpot are available before the drawing, but only at the close of betting is the exact payoff known.
In Euler's time, the technology needed to deliver the sort of uptotheminute information on ticket sales used in the calculation of parimutuel odds was not available. Therefore Euler noted in his letter to Frederick that it was entirely proper for the bank to offer relatively smaller payoffs for the riskier ambo and terno bets in order to insulate itself from calamity. He also observes that if the total number of bets is small, it would be undesirable to have many players choosing the same ambo or terno bets, although it's not clear what sort of practical bookkeeping procedures might have been instituted at that time to avoid these multiple bets.
We note that although Roccolini's proposal did not allow for a gambler to bet on all 5 numbers in the drawing, this sort of bet was permitted in other Genoesestyle lotteries of the 18th century, including the lottery that was eventually instituted in Berlin in 1763. In fact, it was even possible in the French Royal lottery to bet on all five numbers and the order in which they were drawn. The payoff, at a million to one, made this game irresistibly attractive to the French populace, particularly the lower classes, despite the fact that the true odds are in excess of five billion to one. "There were occasions when mathematicians wrote learned articles demonstrating why people should not play the lottery at those odds," writes Katz [9, p. 598] in describing the French lottery, "but the only mathematicians to whom people paid attention were those who sold surefire methods for picking the winning numbers!"
Euler neither wrote cautionary epistles, nor did he hawk surefire winning strategies. Instead, the Genoese lottery inspired in him a series of mathematical articles concerning some of the trickier questions of combinatorics and probability theory to be tackled in the 18th century. The first glimmer of these deeper thoughts is his observation, towards the end of the letter of September 17, that there is nothing sacred about the numbers 5 and 90; the Genoese lottery may be abstracted to any number n of numbered balls or tickets in the urn, and any number t < n of such tokens chosen in the drawing. He closes by outlining for Frederick an alternate lottery scheme, involving n=100, t=10, where players bet on 1, 2, 3 or 4 numbers.
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

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 24. We summarize these in Table 2, where the kth column represents the results of the corresponding Problem.
p_{k,i}  k = 1  k = 2  k = 3  k = 4  
i=0 





i=1 





i=2 




i=3 



i=4 


Table 2: Summary of Problems 14 
p_{k,i}  k = 1  k = 2  k = 3  k = 4  
i=0 





i=1 





i=2 




i=3 



i=4 


Table 2: Summary of Problems 14 
Euler's reasoning is entirely elementary, and each Problem builds on the results of the previous one. If k=1, the player is choosing just one number from among n. Since t of these will be winning numbers, p_{1,1} = t/n and p_{1,0}=1 p_{1,1}.
To calculate the probabilities when k=2, we can think of the player as choosing the two numbers one at a time. Therefore, there are n1 ways to choose the second number, having made the first choice from among n. This accounts for the denominator n(n1) in all three of the probabilities in the second column of Table 2. The case k=2, i=2 arises from choosing one of the t winning numbers first, and one of the remaining t1 winning numbers next. Similary, in the calculation of p_{2,0}, the player first chooses one of the nt losing numbers, and then chooses one of the remaining nt1 losing numbers. The interesting case is i=1. There are two ways this can happen, either by choosing a winning number followed by a losing number, or by choosing a losing number followed by a winning number. There are t good choices and nt bad choices, so the numerator of p_{2,1} has to be 2t(nt).
Elements of the pattern in Table 2 gradually come into focus quickly. Particularly noteworthy is the presence of binomial coefficients. The details of Euler's solutions of Problems 14 make this identification certain, as the coefficient of every entry in each column, except the first and last, arises a sum of two coefficients in the previous column, thereby obeying precisely the same recursive formula as the entries in Pascal's triangle. For example, one can bet on k=4 numbers, and get i=2 of them correct in two different ways: either by matching 2 of the first 3 numbers bet upon, and then failing to match on the fourth one, or by matching only 1 of the first 3, and then succesfully matching the fourth one. Therefore, the binomial coefficient


In Problem 5, Euler asks for the general form of p_{k,i}. He introduces r=nt and writes out the cases k=1, 2, ¼, 6 in this new notation. Observing the pattern, s_{k,i} is seen to be:

Euler does not use this notation, nor does he explicitly give a formula for the general value of s_{k,i}. His notation is as follows:

In Corollary 1 to Problem 5, Euler observes that the s_{k,i} can be given recursively. We may give an explicit formulation as follows. Let
s_{1,0}=  nt

s_{1,1}=  t








In Problem 6, Euler turns his attention to finding the fair prize for a wager one écu. In this discussion, it is clear that Euler is considering a more elaborate lottery scheme than Roccolini's where, for example, a gambler playing terno must match all three of his k=3 selections. In the "Reflections" paper, Euler will award a prize F_{k,i} if i of the players' k numbers match the t numbers drawn, for any 0 < i £ k.
To do this, Euler simply chooses k positive numbers satisfying a_{k,1} + a_{k,2} + ¼+ a_{k,k} = 1, and award prizes


Absent a notation uniform in k, the discussion of this simple point is surprisingly tedious, and is handled one case at a time, ending at k=5. "It's not likely that we'd need to consider more than 5 numbers," Euler says, "as the prizes would be too exorbitant". Of course, it's precisely these `exorbitant' prizes that make so many contemporary lotteries so very irresistible.
The a_{k,i}s are not uniquely determined, unless k=1. So for k > 1, Euler discusses three possible weighting schemes:



Euler motivates methods 2 and 3 as progressively minimizing the impact of large prizes, corresponding to large values of i, on the bank. Curiously, he gives neither formulas nor even explanations for these weights, but simply tabulates the coefficients up to the case k=5. I am grateful to Prof. Stephen Bloch of my department for help in solving the riddle of how the coefficients in method 3 were arrived at.
To illustrate these methods, let us compare in Table 3 the values of a_{5,i} in the three cases.
Method  i=1  i=2  i=3  i=4  i=5  
1 






2 






3 






Table 3: coefficients for k=5 
The question Euler examines in Problem 7 is that of determining a fair rate of return for the bank. Euler mentions that the lottery must hold back something in order to cover its expenses, and observes that if the lottery is being used to finance important state needs, then further discounts on the prizes above and beyond what he recommends may be needed. He suggests that on prizes with only one number matched, the bank should hold back be no more than 10% of the revenue, as a greater discount "would be too obvious and disgust the participants". Thus, 90% of F_{k,1} should be returned to the gambler for each k.
On the other hand, larger profit margins for larger values of i are both justified by the risk, and "will hardly be noticed, given that few people are in a position to calculate the fair value." Accordingly, he suggests returning 80%, 70%, 60% and 50% of F_{k,i} to the player when i=2,3, 4 and 5, respectively. These profit margins of 10%, 20%, and so on, are far more modest than the 31%, 49% and 71% margins spelled out in Roccolini's proposal (see the final column of Table 1).
As with Problem 6, Euler's recommendations in Problem 7 do not have the force of mathematical proof behind them; they amount to little more than the recommendations and educated guesses of a respected scholar.
It is interesting to compare Euler's recommendations to the rules used in a modern lottery, where the parimutuel system insulates the bank from any real risk. In such a situation, prize money is concentrated on the highest values of i, rather than the lowest, because large jackpots attract players to the game. Let's return to the case of New York State, where t=6, and k=6 is the only choice that players have. The first difference we notice is that the bank's cut is a uniform 62% across the board, rather than being graduated according to the size of i. Secondly, the weights are dramatically shifted in favor of the highest value of i, in stark contrast to Euler's second and third method of weighting. Here are the values of a_{6,i} used in the New York State Lotto[]; the figures don't add up to 100% because 7.25% of the prize money goes to those who win the second prize by matching the "bonus number," as described in section 2.
i=1  i=2  i=3  i=4  i=5  i=6 
0  0  6.00%  6.25%  5.50%  75.00% 
Table 4: coefficients for the NYS Lotto 
Having determined the bank's cut and the weighting of prize money, all that remains for Euler is to plug in values of n and t, and to calculate the size of the prizes under each of the three methods. In Problem 8 he does this in the canonical case of n=90 and t=5. Table 5 contains these recommended prizes.
In Problem 9, he uses the values n=100 and t=9, curiously close to the case n=100 and t=10 which he outlined for Frederick in 1749. Once again, he calculates the prize money under all 3 methods for each value of k £5.
k  i  Method 1  Method 2  Method 3  
1  1  16  16  16  
2  2  160  106  64  
1  4 



3  3  2,741  1,174  513  
2  36  47  41  
1  2 



4  4  76,655  20,441  7,130  
3  526  561  391  
2  14 

24  
1  1 



5  5  4,394,927  708,859  207,307  
4  12,409  10,007  5,853  
3  172  278  243  
2  7 



1 


1  
Table 5: Prizes for the canonical lottery 
Here are a few classroom activities which might be incorporated into a lesson plan on lotteries and probability theory. Depending on the approach you take, a presentation of Euler's methods for calculating lottery properties could make for a good topic for any of the grades 914.


Euler wrote a total four papers on the Genoese lottery, as well as a fifth one [13] on an entirely different type of lottery, which was also proposed to Frederick II as a way to raise money, this time by a Dutchman named van Griethouse.
The second of the four Genoese lottery papers, called "On the Probability of Sequences in the Genoese Lottery" [14], was read to the Berlin Academy of Sciences in 1765 and published in the Mémoires of the academy two years later. In this memoir, Euler considers the probability that sequences, or runs of consecutive numbers, will appear among the numbers drawn in a Genoese style lottery. If, for example, the numbers 7, 8, 25, 26 and 27 are drawn, 7 and 8 constitute a sequence of two whereas 25, 26 and 27 constitute a sequence of three. It was not possible to bet on the occurrence of sequences in the Berlin lottery, so we must assume that this is simply a mathematical puzzle which Euler found appealing and worthy of his attention.
After leaving Berlin in 1766, Euler wrote two more memoirs concerning matters of probability theory arising from the Genoese lottery. Both of these papers [15, 16] concerned the number of distinct integers between 1 and n which would be drawn over the course of many repetitions of the lottery. The first of these, "On the Solution of Difficult Questions in the Calculus of Probability" was presented to the Academy of St. Petersburg on October 8, 1781 but published in 1785, two years after Euler's death. Euler begins with the observation that it would be convenient to have a special symbol for the binomial coefficient, and so he defines:

The final paper, entitled "Analysis of a Problem in the Calculus of Probability," was published posthumously in 1862. It is shorter than his other other three papers on the Genoese lottery, and reads more tersely when compared with the lucid technical prose of those three. It was probably a preliminary draft that Euler never managed to put into final form.
All in all, Euler's writings on lotteries comprise but a tiny fraction of his total mathematical output, but they stand as a testament to his love of puzzles and mathematical recreations.
Department of Mathematics and Computer Science
Adelphi University
Garden City, NY 11530