Probability is a subtle concept that is typically defined in terms of the relative frequency of occurrence of an event, or the degree of belief that the event will occur (or that a statement is true, etc.).

Elementary probability
Sometimes introduced in primary school mathematics classes, the first serious study of the subject is usually undertaken in secondary school or university level courses
Probability theory
Formal, axiomatic study at the university level, based on real analysis and especially measure theory.

Major topics

• Interpretations of probability
• Frequentist approach (based on aleatory probability, empirical probability)
• Bayesian approach (based on epistemic probabilty)
• Definitions of probability
• Probability (relative frequency)
• Axioms of probability (or Probability axioms, a.k.a. Kolmogorov axioms)
• Cox's theorem
• Confidence and certainty
• Odds, expected value and outcome, fair and unfair games
• Properties of probability
• Discrete probability versus continuous probability
• Weighted probability
• Probability distribution functions
• Probability density functions
• Cumulative distribution functions
• Conditional probability and mutually exclusive events
• Classical probability and statistics
• Probability and statistics

History

The scientific study of probability is a modern development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions arose much later. There are reasons for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues are still obscured by the superstitions of gamblers.[1]

According to Richard Jeffrey, "Before the middle of the seventeenth century, the term 'probable' (Latin probabilis) meant approvable, and was applied in that sense, univocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances."[2] However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.[3]

The earliest known forms of probability and statistics were developed by Middle Eastern mathematicians studying cryptography between the 8th and 13th centuries. Al-Khalil (717–786) wrote the Book of Cryptographic Messages which contains the first use of permutations and combinations to list all possible Arabic words with and without vowels. Al-Kindi (801–873) made the earliest known use of statistical inference in his work on cryptanalysis and frequency analysis. An important contribution of Ibn Adlan (1187–1268) was on sample size for use of frequency analysis.[4]

The sixteenth-century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes[5]). Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject.[6] Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics.[7] See Ian Hacking's The Emergence of Probability[8] and James Franklin's The Science of Conjecture[9] for histories of the early development of the very concept of mathematical probability.

The theory of errors may be traced back to Roger Cotes's Opera Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation.[10] The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that certain assignable limits define the range of all errors. Simpson also discusses continuous errors and describes a probability curve.

The first two laws of error that were proposed both originated with Pierre-Simon Laplace. The first law was published in 1774, and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error—disregarding sign. The second law of error was proposed in 1778 by Laplace, and stated that the frequency of the error is an exponential function of the square of the error.[11] The second law of error is called the normal distribution or the Gauss law. "It is difficult historically to attribute that law to Gauss, who in spite of his well-known precocity had probably not made this discovery before he was two years old."[11]

Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.

Adrien-Marie Legendre (1805) developed the method of least squares, and introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets).[12] In ignorance of Legendre's contribution, an Irish-American writer, Robert Adrain, editor of "The Analyst" (1808), first deduced the law of facility of error,

${\displaystyle \phi(x) = ce^{-h^2 x^2},}$

where ${\displaystyle h}$ is a constant depending on precision of observation, and ${\displaystyle c}$ is a scale factor ensuring that the area under the curve equals 1. Gauss gave the first proof that seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823), James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), W.F. Donkin (1844, 1856), and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875).

In the nineteenth century, authors on the general theory included Laplace, Sylvestre Lacroix (1816), Littrow (1833), Adolphe Quetelet (1853), Richard Dedekind (1860), Helmert (1872), Hermann Laurent (1873), Liagre, Didion and Karl Pearson. Augustus De Morgan and George Boole improved the exposition of the theory.

In 1906, Andrey Markov introduced[13] the notion of Markov chains, which played an important role in stochastic processes theory and its applications. The modern theory of probability based on the measure theory was developed by Andrey Kolmogorov in 1931.[14]

On the geometric side, contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).[15] See integral geometry for more info.

References

1. Freund, John. (1973) Introduction to Probability. Dickenson ISBN 978-0-8221-0078-2 (p. 1)
2. Jeffrey, R.C., Probability and the Art of Judgment, Cambridge University Press. (1992). pp. 54–55 . ISBN 0-521-39459-7
3. Franklin, J. (2001) The Science of Conjecture: Evidence and Probability Before Pascal, Johns Hopkins University Press. (pp. 22, 113, 127)
4. Broemeling, Lyle D. (1 November 2011). "An Account of Early Statistical Inference in Arab Cryptology". The American Statistician 65 (4): 255–257. doi:10.1198/tas.2011.10191.
5. Some laws and problems in classical probability and how Cardano anticipated them Gorrochum, P. Chance magazine 2012
6. Abrams, William, A Brief History of Probability, Second Moment, retrieved 2008-05-23
7. Ivancevic, Vladimir G.; Ivancevic, Tijana T. (2008). Quantum leap : from Dirac and Feynman, across the universe, to human body and mind. Singapore ; Hackensack, NJ: World Scientific. p. 16. ISBN 978-981-281-927-7.
8. Hacking, I. (2006) The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference, Cambridge University Press, ISBN 978-0-521-68557-3 Template:Page needed
9. Franklin, James (2001). The Science of Conjecture: Evidence and Probability Before Pascal. Johns Hopkins University Press. ISBN 978-0-8018-6569-5.
10. Shoesmith, Eddie (November 1985). "Thomas Simpson and the arithmetic mean" (in en). Historia Mathematica 12 (4): 352–355. doi:10.1016/0315-0860(85)90044-8.
11. Wilson EB (1923) "First and second laws of error". Journal of the American Statistical Association, 18, 143
12. Seneta, Eugene William. ""Adrien-Marie Legendre" (version 9)". StatProb: The Encyclopedia Sponsored by Statistics and Probability Societies. Archived from the original on 3 February 2016. Retrieved 27 January 2016.
13. Weber, Richard. "Markov Chains" (PDF). Statistical Laboratory. University of Cambridge.
14. Vitanyi, Paul M.B. (1988). "Andrei Nikolaevich Kolmogorov". CWI Quarterly (1): 3–18.
15. Wilcox, Rand R. (10 May 2016). Understanding and applying basic statistical methods using R. Hoboken, New Jersey. ISBN 978-1-119-06140-3. OCLC 949759319.