Classical theory of probability

Theory generally attributed to French mathematician and astronomer Pierre-Simon, Marquis de Laplace (1749-1827) in his Essai philosophique sur les probability (1820).

It says that the probability of an occurrence in a given situation is the proportion, among all possible outcomes, of those outcomes that include the given occurrence.

The main difficulty lies in dividing up the alternatives so as to ensure that they are equiprobable, for which purpose Laplace appealed to the controversial principle of indifference.

A related difficulty is that the theory seems to apply to at best a limited range of rather artificial cases, such as those involving throws of dice.

Source:
H E Kyburg, Probability and Inductive Logic (1970), ch. 3

History

As a mathematical subject, the theory of probability arose very late—as compared to geometry for example—despite the fact that we have prehistoric evidence of man playing with dice from cultures from all over the world.^[3] One of the earliest writers on probability was Gerolamo Cardano. He perhaps produced the earliest known definition of classical probability.^[4]

The sustained development of probability began in the year 1654 when Blaise Pascal had some correspondence with his father’s friend Pierre de Fermat about two problems concerning games of chance he had heard from the Chevalier de Méré earlier the same year, whom Pascal happened to accompany during a trip. One problem was the so-called problem of points, a classic problem already then (treated by Luca Pacioli as early as 1494,^[5] and even earlier in an anonymous manuscript in 1400^[5]), dealing with the question how to split the money at stake in a fair way when the game at hand is interrupted half-way through. The other problem was one about a mathematical rule of thumb that seemed not to hold when extending a game of dice from using one die to two dice. This last problem, or paradox, was the discovery of Méré himself and showed, according to him, how dangerous it was to apply mathematics to reality.^[5][6] They discussed other mathematical-philosophical issues and paradoxes as well during the trip that Méré thought was strengthening his general philosophical view.

Pascal, in disagreement with Méré’s view of mathematics as something beautiful and flawless but poorly connected to reality, determined to prove Méré wrong by solving these two problems within pure mathematics. When he learned that Fermat, already recognized as a distinguished mathematician, had reached the same conclusions, he was convinced they had solved the problems conclusively. This correspondence circulated among other scholars at the time, in particular, to Huygens, Roberval and indirectly Caramuel,^[5] and marks the starting point for when mathematicians in general began to study problems from games of chance. The correspondence did not mention “probability”; It focused on fair prices.^[7]

Half a century later, Jacob Bernoulli showed a sophisticated grasp of probability. He showed facility with permutations and combinations, discussed the concept of probability with examples beyond the classical definition (such as personal, judicial and financial decisions) and showed that probabilities could be estimated by repeated trials with uncertainty diminished as the number of trials increased.^[7][8]

The 1765 volume of Diderot and d’Alembert’s classic Encyclopédie contains a lengthy discussion of probability and summary of knowledge up to that time. A distinction is made between probabilities “drawn from the consideration of nature itself” (physical) and probabilities “founded only on the experience in the past which can make us confidently draw conclusions for the future” (evidential).^[9]

The source of a clear and lasting definition of probability was Laplace. As late as 1814 he stated:

The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible.

— Pierre-Simon Laplace, A Philosophical Essay on Probabilities^[10]

This description is what would ultimately provide the classical definition of probability. Laplace published several editions of multiple documents (technical and a popularization) on probability over a half-century span. Many of his predecessors (Cardano, Bernoulli, Bayes) published a single document posthumously.

Criticism

The classical definition of probability assigns equal probabilities to events based on physical symmetry which is natural for coins, cards and dice.

• Some mathematicians object that the definition is circular.^[11] The probability for a “fair” coin is… A “fair” coin is defined by a probability of…
• The definition is very limited. It says nothing about cases where no physical symmetry exists. Insurance premiums, for example, can only be rationally priced by measured rates of loss.
• It is not trivial to justify the principle of indifference except in the simplest and most idealized of cases (an extension of the problem limited definition). Coins are not truly symmetric. Can we assign equal probabilities to each side? Can we assign equal probabilities to any real world experience?

However limiting, the definition is accompanied with substantial confidence. A casino which observes a marked departure from classical probability is confident that its assumptions have been violated (somebody is cheating).^{[citation needed][disputed – discuss]} Much of the mathematics of probability was developed on the basis of this simplistic definition. Alternative interpretations of probability (for example frequentist and subjective) also have problems.

Mathematical probability theory deals in abstractions, avoiding the limitations and philosophical complications of any probability interpretation.

References

• Pierre-Simon de Laplace. Théorie analytique des probabilités. Paris: Courcier Imprimeur, 1812.
• Pierre-Simon de Laplace. Essai philosophique sur les probabilités, 3rd edition. Paris: Courcier Imprimeur, 1816.
• Pierre-Simon de Laplace. Philosophical essay on probabilities. New York: Springer-Verlag, 1995. (Translated by A.I. Dale from the fifth French edition, 1825. Extensive notes.)