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were unpublished and hence cannot be cited as evidence that the problem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 Trattoto di Aritmetica 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.