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Raising Public AwarenessHe points out that one can use his formula to determinethe value of a for which the probability of landingon a single board is fifty percent: “Il y a donc une certainelargeur de la planche qui rendroit le pari ou le jeuégal.” 22. This research was published – in a more extensive version– in 1777, in his Histoire naturelle. It will now becomeeven clearer that the principal motivation for hisinvestigations was in calculating odds for gamblers:Je suppose que, dans une chambre dont le parquet estsimplement divisé par des points parallèles, on jette enl’air une baguette, et que l’un des joueurs parie que labaguette ne croisera aucune des parallèles du parquet,et que l’autre au contraire parie que la baguette croiseraquelques-unes des ces parallèles; on demande lesort de ces deux joueurs. (On peut jouer ce jeu sur undamier avec une aiguille à coudre ou une épingle sanstête.) 33. Buffon later carried out experiments related to theSt. Petersburg paradox. This involves a game in which afair coin is tossed repeatedly until it first lands showingheads. If this occurs at the kth toss, the player wins 2 kducats. Since the expected value of the player’s winningsis infinite, it makes sense that the cost to play the gameshould also be infinite.Buffon describes the game in [3], beginning on page394, and on page 399, one learns that he has made somerelevant experiments:J’ai donc fait deux mille quarante-huit expériences surcette question, c’est-à-dire j’ai joué deux mille quarante-huitfois ce jeu, en faisant jeter la pièce par unenfant. 44. Laplace took up Buffon’s needle problem and statedexplicitly that one could use the theoretical calculationsto determine an experimental approximation of p. Afterdetermining the probability that a line would be crossed,he writes:Si l’on projette un grand nombre de fois ce cylindre,le rapport du nombre de fois où le cylindre rencontreral’une des divisions du plan au nombre total desprojections sera, à très peu près, la valeur de 4r/(a p),2Freely translated: “There is, therefore, a certain width ofboard for which the wager – that is, the game – is fair.”3Freely translated: “I assume that within a room in which theparquet is simply divided by parallel points, one tosses a stickinto the air, and one player wagers that the the stick will notcross any of the parallels in the parquet, while the other playerwagers that it will cross one of the parallels. One asks forthe odds for each player. (One could also play this game on acheckerboard with knitting needles or headless pins.)” (See[3], p. 411ff.)4 Freely translated: I have carried out 2048 experiments withrespect to this question, that is, I have played this game 2048times, making use of a child to toss the coin.”ce qui fera connaître la valeur de la circonférence2p. 5 [9]5. Buffon’s needle problem seems to have sparked interestin actual experiments from the mid-19th centuryonwards. Apparently the first documented one was performedin 1850 by Rudolf Wolf [13], then a professorat the University of Bern. Augustus de Morgan refersin 1859 ([11], pp. 283–4) that a certain Ambrose Smithperformed the experiment in 1855 with 3204 trials and astudent of his with 600 trials.How reasonable are such experiments? The theory saysthat only results of the following type are to be expected.If one tosses the stick n times and hits a line k times, usingk/n in calculating an approximation to p, then with someprobability p, the result that one obtains is within somevalue e of p. Here one may choose a value of p (close to1) and a value of e (small) and one can then determinea suitable value of n, for example using Chebyshev’s inequality.Unfortunately, for moderately large p and notvery small e, the required n is astronomically large andconvergence exceedingly slow. Buffon’s method is thereforenot well suited to obtaining information about thedigits of the number p.It is worth noting in this connection the experimentsreported in 1901 of one Lazzerini, who maintained thatafter throwing 3408 sticks, he had obtained a value of paccurate to six decimal places ([10], p. 120). Papers byGridgeman [5] and Badger [1] refute this claim as extremelyunlikely and probably due to data manipulation.In notable contrast to the historically verified evidence,there remains the impression that Buffon explicitly hadin mind a determination of an approximate value of pby means of an “experiment” and that he in fact carriedit out:1. The Buffon problem is treated regularly in textbookson stochastic theory. In books on this subject in bothEnglish and German, we have found not a single one inwhich the least doubt is cast on the statement that “Buffonwished to calculate an approximation to p with hisexperiment”. (This is true, alas, of the textbook ElementareStochastik of the first author.)2. These textbook authors find themselves in good company,for even in books on the history of the theory ofprobability, it is maintained, without citing any sources,that Buffon performed such experiments. Here are twoexamples:- “It was originally performed with a needle” ([7], p.75).5Freely translated: If one tosses this cylinder with great frequency,then the quotient of the total number of throws forwhich the stick lands on one of the divisions of the plane andthe total number of throws will have approximately the value4r = (a p) which will suffice to determine the value of the circumference2p.”48 EMS Newsletter June 2014