# The Division of Stakes or The Problem of Points

The theory of probability properly begins with the correspondence of Pascal and Fermat. The Chevalier de Méré had proposed to Pascal two problems for solution, one concerning the advantage to a gamester of a wager on the outcome of certain casts of dice and the other concerning the division of stakes in a prematurely terminated game among equally skilled players. This latter problem is now called the Problem of Points.

The Division of Stakes problem, however, occurs much earlier in the literature. The treatments by Pacioli, Cardano and Tartaglia are well-known. However, new traces of the problem predating these have come to light in several Italian manuscripts. One must be reminded that these manuscripts were unpublished and hence cannot be cited as evidence that the problem was introduced to a more general public at that time. Nonetheless it is certain that the problem was known at least a century before the first printed book to contain an example of it. The relevant literature prior to Pascal and Fermat consists of the following.

• In 1985, Laura Toti Rigatelli called attention to a commercial arithmetic (Codex Magliabechiano CL. XI. 120 of the National Library of Florence) by an anonymous author. This has been dated to somewhere near the end of the fourteenth century or beginning of the fifteenth, that is, around the year 1400.
• An anonymous undated manuscript (Urb. lat. 291 of the Vatican Apostolic Library) contains a passage on the division of stakes. The work was discovered by Raffaella Franci.  It is certainly earlier than 1455 and most likely dates from the beginning of the fifteenth century. It is remarkable that the analysis is entirely correct and that it predates Pascal and Fermat by nearly 250 years.
• Codex L. VI. 45 of Filippo Calandri.
• Fra Luca Pacioli, famous for having created the system of double entry bookkeeping, considered the problem in his Summa de Arithmetica, Geometria, Proportioni et Proportionalita printed in Venice 1494.
• Gerolamo Cardano leveled a severe criticism of Pacioli in his Practica arithmetice et mensurandi singularis of 1539.
• Niccoló Tartaglia noted that Pacioli's solution violated common sense and presented his own in the General Trattato di numeri e misure printed at Venice in 1556.
• Gio. Francesco Peverone treats the problem in Due brevi e facili trattati, il primo di Arithmetica, l'altro di Geometria printed at Lyon in 1558.
• Guillaume Gosselin of Caen translated the of Tartaglia into French. This was printed at Paris in 1578. The link is to a 1613 printing.
• Francesco Pagani of Bagnacavallo, Arithmetica prattica utilissima, artificiosamente ordinata printed in 1591 at Ferrara.
• Lorenzo Forestani Pratica d'Arithmetica e Geometria 1602

During the summer and fall of 1654, Pascal and Fermat exchanged solutions to this problem.

Christiaan Huygens incorporated this problem in his De ratiociniis in ludo aleae, a short work, published in 1656/7, consisting of fourteen propositions with proofs and five exercises. Problems IV-IX concern the Problem of Points.

Jakob Bernoulli wrote an extensive commentary on the De ratio which forms the first part of the Ars Conjectandi. Consult pages 16 to 19. Although written prior to 1700, the publication of this work was delayed until 1713.

In the meantime, Pierre Montmort conceived his Essay d'analyse sur les jeux de hazard, published in 1708. Its greatly expanded second edition also appeared in 1713. Discussion of the Problem of Points, together with a reprinting of the letter of Pascal to Fermat dated 29 July 1654, appears on pages 232-248.

The Doctrine of Chances is Moivre's most famous work. It originated in a short paper called "De Mensura Sortis, seu, de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus," Philos. Trans. R. Soc. London 27, 213-264, (1711) for which a modern translation into English has been made by B. McClintock, Int. Stat. Rev. 52, 229-262 (1984). The Doctrine of Chances first appeared in 1718. Later editions were published in 1738 and 1756. Consult page 18 and Proposition VI. Further researches related to the Problem of Points are in the Miscellanea Analytica, 1730.

Nicole contributed two papers in the Hist. de l'Acad...Paris, 1730, pages 45-56 and 331-344. The first is "Examen et resolution de quelques questions sur les jeux" and "Methode pour déterminer le sort de tant de joueurs que l'on voudra, & l'avantage que les uns ont sur les autres, lorsqu'ils jouent à qui gagnera le plus de parties dans un numbre de parties déterminé."

We find a treatment of the problem in Lagrange, "Recherches sur les suites récurrentes dont les termes varient de plusiers manieres différentes, ou sur l'integration des équations linéares aux différences finies et partielles; et sur l'usage de ces équations dans la théorie des hasards." Nouveaux Mémoires de l'Académie royale des Sciences et Belles-Lettres de Berlin, 1775.1. Consult Problems III and IV.

Trembley essentially repeats de Moivre in "Disquisitio Elementaris circa Calculum Probabilium," Commentationes Societas Regiae Scientarum Gottingensis, Vol. XIII, 1793-4, p. 99-136. Consult problems III-V.

Finally, P.S. Laplace  considers the problem in "Mémoire sur la probabilité des causes par les événemens," Savants étranges 6, 1774, p. 621-656. Oeuvres 8, p. 27-65 and in "Recherches, sur l'integration des Équations differentielles aux différences finies, & sur leur usage dans la théorie des hasards." Savants étranges, 1773 (1776) p. ~113-163. Oeuvres 8, p. 69-197. The first has been translated by Stephen Stigler in "Laplace's 1774 Memoir on Inverse Probability," Statistical Science, Vol. 1, Issue 3 (Aug. 1986) 359-363 and "Memoir on the Probability of the Cause of Events," in the same issue, pp. 364-378. Consult Problems XIV and XV in the second.