History of probability
Encyclopedia
Probability
has a dual aspect: on the one hand the probability or likelihood of hypotheses given the evidence for them, and on the other hand the behavior of stochastic processes such as the throwing of dice or coins. The study of the former is historically older in, for example, the law of evidence, while the mathematical treatment of dice began with the work of Pascal
and Fermat
in
the 1650s.
Probability
is distinguished from statistics
. (See History of Statistics
). While statistics deals with data and inferences from it, (stochastic) probability deals with the stochastic (random) processes which lie behind data or outcomes.
probabilis and verisimilis, deriving from Cicero
and generally applied to an opinion to mean plausible or generally approved.
s and half-proof
to deal with the uncertainties of evidence in court.
In Renaissance
times, betting was discussed in terms of odds such as "ten to one" and maritime insurance
premiums were estimated based on intuitive risks, but there was no theory on how to calculate such odds or premiums.
The mathematical methods of probability arose in the correspondence of Pierre de Fermat
and Blaise Pascal
(1654) on such questions as the fair division of the stake in an interrupted game of chance. Christiaan Huygens (1657) gave a comprehensive treatment of the subject.
(posthumous, 1713) and Abraham de Moivre
's The Doctrine of Chances
(1718) put probability on a sound mathematical footing, showing how to calculate a wide range of complex probabilities. Bernoulli proved a version of the fundamental law of large numbers
, which states that in a large number of trials, the average of the outcomes is likely to be very close to the expected value - for example, in 1000 throws of a fair coin, it is likely that there are close to 500 heads (and the larger the number of throws, the closer to half-and-half the proportion is likely to be).
's determination of the orbit of Ceres
from a few observations. The theory of errors used the method of least squares to correct error-prone observations, especially in astronomy, based on the assumption of a normal distribution of errors to determine the most likely true value.
Towards the end of the nineteenth century, a major success of explanation in terms of probabilities was the Statistical mechanics
of Ludwig Boltzmann
and J. Willard Gibbs which explained properties of gases such as temperature in terms of the random motions of large numbers of particles.
The field of the history of probability itself was established by Isaac Todhunter
's monumental History of the Mathematical Theory of Probability from the Time of Pascal to that of Lagrange (1865).
of R. A. Fisher
and Jerzy Neyman
, which is now widely applied in biological and psychological experiments and in clinical trials of drugs. A hypothesis, for example that a drug is usually effective, gives rise to a probability distribution
that would be observed if the hypothesis is true. If observations approximately agree with the hypothesis, it is confirmed, if not, the hypothesis is rejected.
The theory of stochastic processes broadened into such areas as Markov process
es and Brownian motion
, the random movement of tiny particles suspended in a fluid. That provided a model for the study of random fluctuations in stock markets, leading to the use of sophisticated probability models in mathematical finance
, including such successes as the widely-used Black–Scholes formula for the valuation of options
.
The twentieth century also saw long-running disputes on the interpretations of probability
. In the mid-century frequentism
was dominant, holding that probability means long-run relative frequency in a large number of trials. At the end of the century there was some revival of the Bayesian
view, according to which the fundamental notion of probability is how well a proposition is supported by the evidence for it.
The mathematical treatment of probabilities, especially when there are infinitely many possible outcomes, was facilitated by Kolmogorov's axioms
(1931).
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...
has a dual aspect: on the one hand the probability or likelihood of hypotheses given the evidence for them, and on the other hand the behavior of stochastic processes such as the throwing of dice or coins. The study of the former is historically older in, for example, the law of evidence, while the mathematical treatment of dice began with the work of 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...
and 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...
in
the 1650s.
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 distinguished from statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
. (See History of Statistics
History of statistics
The history of statistics can be said to start around 1749 although, over time, there have been changes to the interpretation of what the word statistics means. In early times, the meaning was restricted to information about states...
). While statistics deals with data and inferences from it, (stochastic) probability deals with the stochastic (random) processes which lie behind data or outcomes.
Etymology
Probable and likely and their cognates in other modern languages derive from medieval learned LatinMedieval Latin
Medieval Latin was the form of Latin used in the Middle Ages, primarily as a medium of scholarly exchange and as the liturgical language of the medieval Roman Catholic Church, but also as a language of science, literature, law, and administration. Despite the clerical origin of many of its authors,...
probabilis and verisimilis, deriving from Cicero
Cicero
Marcus Tullius Cicero , was a Roman philosopher, statesman, lawyer, political theorist, and Roman constitutionalist. He came from a wealthy municipal family of the equestrian order, and is widely considered one of Rome's greatest orators and prose stylists.He introduced the Romans to the chief...
and generally applied to an opinion to mean plausible or generally approved.
Origins
Ancient and medieval law of evidence developed a grading of degrees of proof, probabilities, presumptionPresumption
In the law of evidence, a presumption of a particular fact can be made without the aid of proof in some situations. The types of presumption includes a rebuttable discretionary presumption, a rebuttable mandatory presumption, and an irrebuttable or conclusive presumption. The invocation of a...
s and half-proof
Half-proof
Half-proof , was a concept of medieval Roman law, describing a level of evidence between mere suspicion and the full proof needed to convict someone of a crime...
to deal with the uncertainties of evidence in court.
In Renaissance
Renaissance
The Renaissance was a cultural movement that spanned roughly the 14th to the 17th century, beginning in Italy in the Late Middle Ages and later spreading to the rest of Europe. The term is also used more loosely to refer to the historical era, but since the changes of the Renaissance were not...
times, betting was discussed in terms of odds such as "ten to one" and maritime insurance
Insurance
In law and economics, insurance is a form of risk management primarily used to hedge against the risk of a contingent, uncertain loss. Insurance is defined as the equitable transfer of the risk of a loss, from one entity to another, in exchange for payment. An insurer is a company selling the...
premiums were estimated based on intuitive risks, but there was no theory on how to calculate such odds or premiums.
The mathematical methods of probability arose in the correspondence of 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...
and 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...
(1654) on such questions as the fair division of the stake in an interrupted game of chance. Christiaan Huygens (1657) gave a comprehensive treatment of the subject.
Eighteenth century
Jacob Bernoulli's Ars ConjectandiArs Conjectandi
Ars Conjectandi is a combinatorial mathematical paper written by Jakob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, most notably among other combinatorial topics, probability theory: indeed, it is widely regarded as...
(posthumous, 1713) and Abraham de Moivre
Abraham de Moivre
Abraham de Moivre was a French mathematician famous for de Moivre's formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmund Halley, and James Stirling...
's The Doctrine of Chances
The Doctrine of Chances
The Doctrine of Chances was the first textbook on probability theory, written by 18th-century French mathematician Abraham de Moivre and first published in 1718. De Moivre wrote in English because he resided in England at the time, having fled France to escape the persecution of Huguenots...
(1718) put probability on a sound mathematical footing, showing how to calculate a wide range of complex probabilities. Bernoulli proved a version of the fundamental law of large numbers
Law of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...
, which states that in a large number of trials, the average of the outcomes is likely to be very close to the expected value - for example, in 1000 throws of a fair coin, it is likely that there are close to 500 heads (and the larger the number of throws, the closer to half-and-half the proportion is likely to be).
Nineteenth century
The power of probabilistic methods in dealing with uncertainty was shown by GaussCarl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
's determination of the orbit of Ceres
CERES
CERES may refer to:* California Environmental Resources Evaluation System * Centre for European, Russian, and Eurasian Studies of the University of Toronto...
from a few observations. The theory of errors used the method of least squares to correct error-prone observations, especially in astronomy, based on the assumption of a normal distribution of errors to determine the most likely true value.
Towards the end of the nineteenth century, a major success of explanation in terms of probabilities was the Statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
of Ludwig Boltzmann
Ludwig Boltzmann
Ludwig Eduard Boltzmann was an Austrian physicist famous for his founding contributions in the fields of statistical mechanics and statistical thermodynamics...
and J. Willard Gibbs which explained properties of gases such as temperature in terms of the random motions of large numbers of particles.
The field of the history of probability itself was established by Isaac Todhunter
Isaac Todhunter
Isaac Todhunter FRS , was an English mathematician who is best known today for the books he wrote on mathematics and its history.- Life and work :...
's monumental History of the Mathematical Theory of Probability from the Time of Pascal to that of Lagrange (1865).
Twentieth century
Probability and statistics became closely connected through the work on hypothesis testingStatistical hypothesis testing
A statistical hypothesis test is a method of making decisions using data, whether from a controlled experiment or an observational study . In statistics, a result is called statistically significant if it is unlikely to have occurred by chance alone, according to a pre-determined threshold...
of R. A. Fisher
Ronald Fisher
Sir Ronald Aylmer Fisher FRS was an English statistician, evolutionary biologist, eugenicist and geneticist. Among other things, Fisher is well known for his contributions to statistics by creating Fisher's exact test and Fisher's equation...
and Jerzy Neyman
Jerzy Neyman
Jerzy Neyman , born Jerzy Spława-Neyman, was a Polish American mathematician and statistician who spent most of his professional career at the University of California, Berkeley.-Life and career:...
, which is now widely applied in biological and psychological experiments and in clinical trials of drugs. A hypothesis, for example that a drug is usually effective, gives rise to a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
that would be observed if the hypothesis is true. If observations approximately agree with the hypothesis, it is confirmed, if not, the hypothesis is rejected.
The theory of stochastic processes broadened into such areas as Markov process
Markov process
In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...
es and Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
, the random movement of tiny particles suspended in a fluid. That provided a model for the study of random fluctuations in stock markets, leading to the use of sophisticated probability models in mathematical finance
Mathematical finance
Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...
, including such successes as the widely-used Black–Scholes formula for the valuation of options
Valuation of options
In finance, a price is paid or received for purchasing or selling options. This price can be split into two components.These are:* Intrinsic Value* Time Value-Intrinsic Value:...
.
The twentieth century also saw long-running disputes on the interpretations of probability
Probability interpretations
The word probability has been used in a variety of ways since it was first coined in relation to games of chance. Does probability measure the real, physical tendency of something to occur, or is it just a measure of how strongly one believes it will occur? In answering such questions, we...
. In the mid-century frequentism
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...
was dominant, holding that probability means long-run relative frequency in a large number of trials. At the end of the century there was some revival of the Bayesian
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...
view, according to which the fundamental notion of probability is how well a proposition is supported by the evidence for it.
The mathematical treatment of probabilities, especially when there are infinitely many possible outcomes, was facilitated by Kolmogorov's axioms
Probability axioms
In probability theory, the probability P of some event E, denoted P, is usually defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov, which are described below....
(1931).
External links
- JEHPS: Recent publications in the history of probability and statistics
- Electronic Journ@l for History of Probability and Statistics/Journ@l Electronique d'Histoire des Probabilitéet de la Statistique
- Figures from the History of Probability and Statistics (Univ. of Southampton)
- Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)
- Earliest Uses of Symbols in Probability and Statistics on Earliest Uses of Various Mathematical Symbols