Classical definition of probability
Encyclopedia
The classical definition of probability
is identified with the works of Pierre-Simon Laplace
. As stated in his Théorie analytique des probabilités,
This definition is essentially a consequence of the principle of indifference
. If elementary events are assigned equal probabilities, then the probability of a disjunction of elementary events is just the number of events in the disjunction divided by the total number of elementary events.
The classical definition of probability was called into question by several writers of the nineteenth century, including John Venn
and George Boole
. The frequentist definition of probability
became widely accepted as a result of their criticism, and especially through the works of R.A. Fisher. The classical definition enjoyed a revival of sorts due to the general interest in Bayesian probability
.
for example—despite the fact that we have prehistoric evidence of man playing with dice from cultures from all over the world. In fact we have the exact year when it was born; in the year 1654 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), 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. 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, and marks the starting point for when mathematicians in general began to study problems from games of chance.
This does not mean that Pascal and Fermat had a clear concept of probability, nor that they made the first correct calculations concerning games of chance. No clear distinction had yet been made between probabilities and expected value
s. The first person known to have seen the need for a clear definition of probability was Laplace. As late as 1814 he stated:
This description is what would ultimately provide the classical definition of probability.
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
is identified with the works of Pierre-Simon Laplace
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...
. As stated in his Théorie analytique des probabilités,
- The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible.
This definition is essentially a consequence of the principle of indifference
Principle of indifference
The principle of indifference is a rule for assigning epistemic probabilities.Suppose that there are n > 1 mutually exclusive and collectively exhaustive possibilities....
. If elementary events are assigned equal probabilities, then the probability of a disjunction of elementary events is just the number of events in the disjunction divided by the total number of elementary events.
The classical definition of probability was called into question by several writers of the nineteenth century, including John Venn
John Venn
Donald A. Venn FRS , was a British logician and philosopher. He is famous for introducing the Venn diagram, which is used in many fields, including set theory, probability, logic, statistics, and computer science....
and George Boole
George Boole
George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...
. The frequentist definition of probability
Frequency probability
Frequency probability is the interpretation of probability that defines an event's probability as the limit of its relative frequency in a large number of trials. The development of the frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the...
became widely accepted as a result of their criticism, and especially through the works of R.A. Fisher. The classical definition enjoyed a revival of sorts due to the general interest in Bayesian probability
Bayesian probability
Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...
.
History
As a mathematical subject the theory of probability arose very late—as compared to geometryGeometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
for example—despite the fact that we have prehistoric evidence of man playing with dice from cultures from all over the world. In fact we have the exact year when it was born; in the year 1654 Blaise Pascal
Blaise Pascal
Blaise Pascal , was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen...
had some correspondence with his father's friend Pierre de Fermat
Pierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
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
Problem of points
The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory. One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal to the first explicit reasoning about what today is...
, a classic problem already then (treated by Luca Pacioli
Luca Pacioli
Fra Luca Bartolomeo de Pacioli was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and seminal contributor to the field now known as accounting...
as early as 1494), 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. 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, and marks the starting point for when mathematicians in general began to study problems from games of chance.
This does not mean that Pascal and Fermat had a clear concept of probability, nor that they made the first correct calculations concerning games of chance. No clear distinction had yet been made between probabilities and expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
s. The first person known to have seen the need for a clear definition of probability was Laplace. As late as 1814 he stated:
This description is what would ultimately provide the classical definition of probability.