Pascal and Fermat Devise the Theory of Probability

Probability theory has become one of the most widely applied branches of mathematics. Two mathematicians, Blaise Pascal and Pierre de Fermat, responded to an inquiry about how to split the stakes from a game, laying the foundations for the subject. Shortly thereafter, the first textbook on the subject was written by Dutch mathematician Christiaan Huygens, who had been thinking about the same issues.

Summary of Event

Probability is the branch of mathematics that assesses how likely certain outcomes are when an experiment is performed. It entered the mathematical literature in the form of questions about games of dice, especially in the work of Gerolamo Cardano Cardano, Gerlamo (1501-1576). These questions did not seem to have attracted much attention elsewhere, and Cardano’s own work suffered from errors. It was not clear at the time how best to define even the basic notions on the basis of which to perform calculations involving events of chance. [kw]Pascal and Fermat Devise the Theory of Probability (1654)
[kw]Probability, Pascal and Fermat Devise the Theory of (1654)
[kw]Fermat Devise the Theory of Probability, Pascal and (1654)
Mathematics;1654: Pascal and Fermat Devise the Theory of Probability[1790]
Netherlands;1654: Pascal and Fermat Devise the Theory of Probability[1790]
France;1654: Pascal and Fermat Devise the Theory of Probability[1790]
Pascal, Blaise
Fermat, Pierre de

Antoine Gombaud, Méré, chevalier de the chevalier de Méré (1607-1684), was a French nobleman with some claim to having done some mathematical work, although his published remains do not provide any basis for judging the quality of his work. He was also a gambler and had investigated two problems. One was that of how to divide up the stakes in a game of dice when the game had to be broken off before it was finished. The other involved the likelihood of throwing a certain number of sixes in a certain number of throws of dice. De Méré knew how many tosses it would take for there to be more than a 50 percent chance of at least one six showing up. He assumed that if he multiplied that by six, he would have the number of tosses it would take for the likelihood of at least two sixes showing up to be more than 50 percent.

Experience showed that this was incorrect (and it had been the view of Cardano, although de Méré was unfamiliar with his work).

Recognizing that he was out of his depth, de Méré turned to the eminent French mathematician Blaise Pascal. Pascal recognized the interest of the problems that had been proposed and initiated a correspondence in 1654 with perhaps the most accomplished mathematician of the period, Pierre de Fermat. It is arguable that this led two of the world’s greatest mathematicians to spend their time looking at a problem raised in the context of gambling. Both Pascal and Fermat were able to recognize the mathematical issues underlying the problem, and between them they created the theory of probability.

Blaise Pascal.

(Library of Congress)

The nature of their arguments involved a precise analysis of the collection of possible outcomes at each stage of the games being played. Starting with a small number of principles, they could tackle both of the problems raised by de Méré by the use of a process now known as recursion. This involves recognizing at certain stages of the game that the situation is exactly the same as it was at a previous turn and deriving from that recognition an algebraic equation that could be solved easily. Both Pascal and Fermat felt satisfied with the solutions that they obtained, although the absence of various pieces of the correspondence does not provide a basis for always being able to judge the generality of their arguments.

One of the key ingredients to Pascal’s solution was the triangle that bears his name. The triangle starts with a 1 at its apex, has two 1’s in the next row, and continues with 1’s at the ends of each row and interior elements obtained by adding up the two numbers immediately adjacent to it in the previous row. This particular triangle had been known for many years and went back at least to medieval Arabic mathematicians. What Pascal recognized was the way in which the numbers in a given row corresponded to the coefficients in expansions of a binomial expression, such as raising a + b to the nth power. The amount of mathematical ingenuity that Pascal lavished on the triangle was impressive, but more surprising was the extent to which it enabled him thereby to answer questions about probability as well.

Pierre de Fermat.

(Library of Congress)

Fermat’s method of proceeding is less well documented, as is frequently the case with Fermat’s work. His inclination was seldom to produce more than the details asked for in a problem rather than the method of proof. His willingness to calculate at length to enumerate all the possible outcomes of an experiment was the basis for his results, which agreed with those of Pascal.

Pascal had a religious conversion shortly after his correspondence with Fermat and gave up mathematics, to a large extent. He made one further contribution to probability, however, which suggested the wider applications of their work. He framed an argument for belief in God that he suggested would be useful in arguing with those who needed to see everything put in terms of games and gambling. The argument used the idea of expectation and has remained an important contribution to philosophy.

The idea of “expectation” is connected with that of “average,” and the rise of probability in the seventeenth century was perhaps connected with the availability of large quantities of data coming from national governments and other large bodies, such as municipalities. This notion provided the basis for the treatise on probability put together by the Dutch mathematician Christiaan Huygens Huygens, Christiaan (Libellus de ratiociniis in ludo aleae, 1657; The Value of All Chances in Games of Fortune
Value of All Chances in Games of Fortune, The (Huygens) , 1714). It is not clear how familiar he was with Fermat’s and Pascal’s work, but he did write the first systematic treatise on the rudiments of probability. From a simple axiom he derived three theorems, and on the strength of those he explained the solutions to a sequence of problems, relying on the same sort of technique that had been used by Pascal. Where Pascal had used the combinatorial ideas embodied in his triangle, however, Huygens just lumbered through long computations. In a way, Huygens’s work was a step back, but his casting the ideas of probability in a systematic form helped the subject to get something of a foothold among mathematicians.


Until the time of Pascal and Fermat, there had been a tendency to appeal to arguments from inspiration and authority in many spheres. By the middle of the seventeenth century, the continued hostilities between Catholic and Protestant forces had cooled down to confrontations rather than conflict. In such a setting there was a call for the kind of argument that depended on something that could be accepted by both sides. Mathematics provided such a setting, and so there was a call for the ideas of probability in both Protestant and Catholic Europe.

Although the correspondence of Pascal and Fermat was not immediately available to subsequent mathematicians, the treatise by Huygens provided some impetus for further research. By the end of the century, there was an explosion of interest in probability, and a number of treatments of the basis of the subject took the place of Huygens’s original work. Even in the middle of the eighteenth century, however, the leading authority on probability could look back on the subject as having been the creation of Pascal and Fermat. They had not been the first mathematicians to consider questions arising from games of chance, but they were the first to apply enough mathematical systematization to the subject to make sure that they did not fall into the traps that had bedeviled their predecessors and continue to afflict those who assess questions of probability without mathematics.

Further Reading

  • Bernstein, Peter L. Against the Gods: The Remarkable Story of Risk. New York: John Wiley and Sons, 1996. A popular examination of the applications of probability to practical issues. Devotes an entire chapter to Pascal and Fermat.
  • David, Florence N. Games, Gods, and Gambling: The Origins and History of Probability and Statistical Ideas from the Earliest Times to the Newtonian Era. New York: Hafner, 1962. This study looks back to various roots for ideas of probability in Western thought and culminates with the generation after Huygens.
  • Gigerenzer, Gerd, et al. The Empire of Chance: How Probability Changed Science and Everyday Life. New York: Cambridge University Press, 1989. This volume primarily looks at how the ideas of Pascal and Fermat about the foundations of the subject affected their successors.
  • Gonick, Larry, and Woollcott Smith. The Cartoon Guide to Statistics. New York: HarperCollins, 1993. This volume not only explains the basic ideas and vocabulary of statistics but also uses seventeenth century France as part of the background.
  • Hacking, Ian. The Emergence of Probability: A Philosophical Study of Early Ideas About Probability, Induction, and Statistical Inference. Cambridge, England: Cambridge University Press, 1975. This is the most sophisticated philosophical examination of Pascal and Fermat and argues that Pascal’s religious use of probability is more important than his strictly mathematical work on the subject.
  • Hald, Anders. A History of Probability and Statistics and Their Applications Before 1750. New York: John Wiley and Sons, 1990. The most extensive and technically polished examination of Pascal, Fermat, and Huygens.
  • Maistrov, L. E. Probability Theory: A Historical Sketch. Translated and edited by Samuel Kotz. New York: Academic Press, 1974. This work attempts to debunk some of the stories about de Méré as a gambler contributing to the foundation of probability theory.
  • Todhunter, Isaac. A History of the Mathematical Theory of Probability: From the Time of Pascal to That of Laplace. Sterling, Va.: Thoemmes Press, 2001. A reprint of an 1865 account that tries to reconstruct what Fermat and Pascal wrote to each other by filling in the published record.

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