## Casanova’s Lottery [book review]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on January 12, 2023 by xi'an

This “history of a revolutionary game of chance” is the latest book by Stephen Stigler and is indeed of an historical nature, following the French Lottery from its inception as Loterie royale in 1758 to the Loterie Nationale in 1836 (with the intermediate names of Loterie de France, Loterie Nationale, Loterie impériale, Loterie royale reflecting the agitated history of the turn of that Century!).

The incentive for following this State lottery is that it is exceptional by its mathematical foundations. Contrary to other lotteries of the time, it was indeed grounded on the averaging of losses and gains on the long run (for the State). The French (Royal) State thus accepted the possibility of huge losses at some draws since they would be compensated by even larger gains. The reasoning proved most correct since the Loterie went providing as far as 4% of the overall State budget, despite the running costs of maintaining a network of betting places and employees, who had to be mathematically savy in order to compute the exact gains of the winners.This is rather amazing as the understanding of the Law of Large Numbers was quite fresh (on an historical scale) thanks to the considerable advances made by Pascal, Fermat, (Jakob) Bernoulli and a few others. (The book mentions the Encyclopedist and mathematician Jean d’Alembert as being present at the meeting that decided of the creation of the Loterie in 1757.)

One may wonder why Casanova gets the credit for this lottery. In true agreement with Stigler’s Law, it is directly connected with the Genoan lottery and subsequent avatars in some Italian cities, including Casanova’s Venezia. But jack-of-all-trades Casanova was instrumental in selling the notion to the French State, having landed in Paris after a daring flight from the Serenissima’s jails. After succeeding in convincing the King’s officers to launch the scheme crafted by a certain Ranieri (de’) Calzabig—not to be confused with the much maligned Salieri!—who would later collaborate with Gluck on Orfeo ed Eurydice and Alceste, Casanova received a salary from the Loterie administration and further run several betting offices. Until he left Paris for further adventures! Including an attempt to reproduce the lottery in Berlin, where Frederick II proved less receptive than Louis XIV. (Possibly due to Euler’s cautionary advice.) The final sentence of the book stands by its title: “It was indeed Casanova’s lottery” (p.210).

Unsurprisingly, given Stephen’s fascination for Pierre-Simon Laplace, the great man plays a role in the history, first by writing in 1774 one of his earliest papers on a lottery problem, namely the distribution of the number of draws needed for all 90 numbers to appear. His (correct) solution is an alternating sum whose derivation proved a numerical challenge. Thirty years later, Laplace came up with a good and manageable approximation (see Appendix Two). Laplace also contributed to the end of the Loterie by arguing on moral grounds against this “voluntary” tax, along Talleyrand, a fellow in perpetually adapting to the changing political regimes. It is a bit of a surprise to read that this rather profitable venture ended up in 1836, more under bankers’ than moralists´ pressure. (A new national lottery—based on printed tickets rather than bets on results—was created a century later, in 1933 and survived the second World War, with the French Loto appearing in 1974 as a direct successor to Casanova’s lottery.)

The book covers many fascinating aspects, from the daily run of the Loterie, to the various measures (successfully) taken against fraud, to the survival during the Révolution and its extension through (the Napoleonic) Empire, to tests for fairness thanks to numerous data from almanacs, to the behaviour of bettors and the sale of “helping” books. to (Daniel) Bernoulli, Buffon, Condorcet, and Laplace modelling rewards and supporting decreasing marginal utility. Note that there are hardly any mathematical formula, except for an appendix on the probabilities of wins and the returns, as well as Laplace’s (and Legendre’s) derivations. Which makes the book eminently suited for a large audience, the more thanks to Stephen Stigler’s perfect style.

This (paperback) book is also very pleasantly designed by the University of Chicago Press, with a plesant font (Adobe Calson Pro) and a very nice cover involving Laplace undercover, taken from a painting owned by the author. The many reproductions of epoch documents are well-done and easily readable. And, needless to say given the scholarship of Stephen, the reference list is impressive.

The book is testament to the remarkable skills of Stephen who searched for material over thirty years, from Parisian specialised booksellers to French, English, and American archives. He manages to bring into the story a wealth of connections and characters, as for instance Voltaire’s scheme to take advantage of an earlier French State lottery aimed at reimbursing State debtors. (Voltaire actually made a fortune of several millions francs out of this poorly designed lottery.) For my personal instructions, the book also put life to several Métro stations like Pereire and Duverney. But the book‘s contents will prove fascinating way beyond Parisian locals and francophiles. Enjoy!

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE. As appropriate for a book about capitalising on chance beliefs!]

## Mr Meyrowitz’s glasses

Posted in Statistics, University life with tags , , , , , , , on October 23, 2011 by xi'an

Today, I found a site entitled Mr Meyrowitz’s Class that links to my first post on coincidences in lotteries as an example of “fatal error”. This seems to be part of a student’s assignment, apparently for the CollegeBoard programme, with 10 minutes allocated to students to find my “fatal error with decimals and probabilities”… As there is no hint, I wonder where my fatal error stands: I could not find it after those 10 minutes of intense searching and recomputing. Maybe Mr Meyrowitz actually needs new glasses to spot the difference between a 1‰ chance and a 1% chance… (Which actually misled a few other readers of the post.)

Question 6) in this assignment also sounds very much inspired from another of my posts on coincidences in lotteries [although not acknowledged in the assignment] since the question refers to the same original France Soir article in French. The question is however rather vague: “do you suspect him of cheating?” and it shows a lack of knowledge about French loto where cheating is [close to] impossible. It is certainly not recommended as an exercise for beginning students in probability or statistics. [Actually, in my opinion, the whole assignment is poor, being either imprecise, e.g question 7), useless, as for question 4) “Pick one topic that you understand very well and one that you do not understand well” (!), or plain wrong, as for question 2)…]

## another lottery coincidence

Posted in R, Statistics with tags , , , on August 30, 2011 by xi'an

Once again, meaningless figures are published about a man who won the French lottery (Le Loto) for the second time. The reported probability of the event is indeed one chance out of 363 (US) trillions (i.e., billions in the metric system. or 1012)… This number is simply the square of

${49 \choose 5}\times{10 \choose 1} = 19,068,840$

which is the number of possible loto grids. Thus, the probability applies to the event “Mr so-&-so plays a winning grid of Le Loto on May 6, 1995 and a winning grid of Le Loto on July 27, 2011“. But this is not the event that occured: one of the bi-weekly winners of Le Loto won a second time and this was spotted by Le Loto spokepersons. If we take the specific winner for today’s draw, Mrs such-&-such, who played bi-weekly one single grid since the creation of Le Loto in 1976, i.e. about 3640 times, the probability that she won earlier is of the order of

$1-\left(1-\frac{1}{{49\choose 5}\times{10\choose 1}}\right)^{3640}=2\cdot 10^{-4}$.

There are thus two chances in 10 thousands to win again for a given (unigrid) winner, not much indeed, but no billion involved either. Now, this is also the probability that, for a given draw (like today’s draw), one of the 3640 previous winners wins again (assuming they all play only one grid,  play independently from each other, &tc.). Over a given year, i.e. over 104 draws, the probability that there is no second-time winner is thus approximately

$\left(1-\frac{1}{2\cdot10^4}\right)^{104} = 0.98,$

showing that within a year there is a 2% chance to find an earlier winner. Not so extreme, isn’t it?! Therefore, less bound to make the headlines…

Now, the above are rough and conservative calculations. The newspaper articles about the double winner report that the man is playing about 1000 euros a month (this is roughly the minimum wage!), representing the equivalent of 62 grids per draw (again I am simplifying to get the correct order of magnitude). If we repeat the above computations, assuming this man has played 62 grids per draw from the beginning of the game in 1976 till now, the probability that he wins again conditional on the fact that he won once is

$1-\left(1-\frac{62}{{49 \choose 5}\times{10 \choose 1}}\right)^{3640} = 0.012$,

a small but not impossible event. (And again, we consider the probability only for Mr so-&-so, while the event of interest does not.) (I wrote this post before Alex pointed out the four-time lottery winner in Texas, whose “luck” seems more related with the imperfections of the lottery process…)

I also stumbled on this bogus site providing the “probabilities” (based on the binomial distribution, nothing less!) for each digit in Le Loto, no need for further comments. (Even the society that runs Le Loto hints at such practices, by providing the number of consecutive draws a given number has not appeared, with the sole warning “N’oubliez jamais que le hasard ne se contrôle pas“, i.e. “Always keep in mind that chance cannot be controlled“…!)