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Probability is a way of expressing knowledge or belief that an event will occur or has occurred. In mathematics the concept has been given an exact meaning in probability theory, that is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
- Main article: Probability interpretations
The word probability does not have a consistent direct definition. In fact, there are sixteen broad categories of probability interpretations, whose adherents possess different (and sometimes conflicting) views about the fundamental nature of probability:
- Frequentists talk about probabilities only when dealing with experiments that are random and well-defined. The probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment. Frequentists consider probability to be the relative frequency "in the long run" of outcomes.
- Bayesians, however, assign probabilities to any statement whatsoever, even when no random process is involved. Probability, for a Bayesian, is a way to represent an individual's degree of belief in a statement, given the evidence.
The word probability derives from probity, a measure of the authority of a witness in a legal case in Europe, and often correlated with the witness's nobility. In a sense, this differs much from the modern meaning of probability, which, in contrast, is used as a measure of the weight of empirical evidence, and is arrived at from inductive reasoning and statistical inference.
- Further information: History of probability
- Further information: History of statistics
The scientific study of probability is a modern development. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions of use in those problems only arose much later.
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." However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.
Aside from some elementary considerations made by Girolamo Cardano in the 16th century, 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. Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics. See Ian Hacking's The Emergence of Probability and James Franklin's The Science of Conjecture 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. The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that there are certain assignable limits within which all errors may be supposed to fall; continuous errors are discussed and a probability curve is given.
Pierre-Simon Laplace (1774) made the first attempt to deduce a rule for the combination of observations from the principles of the theory of probabilities. He represented the law of probability of errors by a curve , being any error and its probability, and laid down three properties of this curve:
- it is symmetric as to the -axis;
- the -axis is an asymptote, the probability of the error being 0;
- the area enclosed is 1, it being certain that an error exists.
He also gave (1781) a formula for the law of facility of error (a term due to Lagrange, 1774), but one which led to unmanageable equations. Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
The method of least squares is due to Adrien-Marie Legendre (1805), who introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets). 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,
being a constant depending on precision of observation, and a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as John Herschel's (1850). Gauss gave the first proof which 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). Peters's (1856) formula for , the probable error of a single observation, is well known.
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.
On the geometric side (see integral geometry) contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).
- Further information: Probability theory
In mathematics, a probability of an event A is represented by a real number in the range from 0 to 1 and written as P(A), p(A) or Pr(A). An impossible event has a probability of 0, and a certain event has a probability of 1. However, the converses are not always true: probability 0 events are not always impossible, nor probability 1 events certain. The rather subtle distinction between "certain" and "probability 1" is treated at greater length in the article on "almost surely".
The opposite or complement of an event A is the event [not A] (that is, the event of A not occurring); its probability is given by P(not A) = 1 - P(A). As an example, the chance of not rolling a six on a six-sided die is 1 - (chance of rolling a six) = . See Complementary event for a more complete treatment.
If both the events A and B occur on a single performance of an experiment this is called the intersection or joint probability of A and B, denoted as . If two events, A and B are independent then the joint probability is
for example, if two coins are flipped the chance of both being heads is 
If either event A or event B or both events occur on a single performance of an experiment this is called the union of the events A and B denoted as . If two events are mutually exclusive then the probability of either occurring is
For example, the chance of rolling a 1 or 2 on a six-sided die is
If the events are not mutually exclusive then
For example, when drawing a single card at random from a regular deck of cards, the chance of getting a heart or a face card (J,Q,K) (or one that is both) is , because of the 52 cards of a deck 13 are hearts, 12 are face cards, and 3 are both: here the possibilities included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards" but should only be counted once.
Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the probability of A, given B". It is defined by
If then is undefined.
|A or B|
|A and B|
|A given B|
- Main article: Probability theory
Like other theories, the theory of probability is a representation of probabilistic concepts in formal terms—that is, in terms that can be considered separately from their meaning. These formal terms are manipulated by the rules of mathematics and logic, and any results are then interpreted or translated back into the problem domain.
There have been at least two successful attempts to formalize probability, namely the Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation (see probability space), sets are interpreted as events and probability itself as a measure on a class of sets. In Cox's theorem, probability is taken as a primitive (that is, not further analyzed) and the emphasis is on constructing a consistent assignment of probability values to propositions. In both cases, the laws of probability are the same, except for technical details.
There are other methods for quantifying uncertainty, such as the Dempster-Shafer theory or possibility theory, but those are essentially different and not compatible with the laws of probability as they are usually understood.
Two major applications of probability theory in everyday life are in risk assessment and in trade on commodity markets. Governments typically apply probabilistic methods in environmental regulation where it is called "pathway analysis", often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on statistical analyses of their probable effect on the population as a whole.
A good example is the effect of the perceived probability of any widespread Middle East conflict on oil prices - which have ripple effects in the economy as a whole. An assessment by a commodity trader that a war is more likely vs. less likely sends prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are not assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict.
It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has had a profound effect on modern society. Accordingly, it may be of some importance to most citizens to understand how odds and probability assessments are made, and how they contribute to reputations and to decisions, especially in a democracy.
Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, utilize reliability theory in the design of the product in order to reduce the probability of failure. The probability of failure may be closely associated with the product's warranty.
Relation to randomness
- Main article: Randomness
In a deterministic universe, based on Newtonian concepts, there is no probability if all conditions are known. In the case of a roulette wheel, if the force of the hand and the period of that force are known, then the number on which the ball will stop would be a certainty. Of course, this also assumes knowledge of inertia and friction of the wheel, weight, smoothness and roundness of the ball, variations in hand speed during the turning and so forth. A probabilistic description can thus be more useful than Newtonian mechanics for analyzing the pattern of outcomes of repeated rolls of roulette wheel. Physicists face the same situation in kinetic theory of gases, where the system, while deterministic in principle, is so complex (with the number of molecules typically the order of magnitude of Avogadro constant 6.02·1023) that only statistical description of its properties is feasible.
A revolutionary discovery of 20th century physics was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of quantum mechanics. The wave function itself evolves deterministically as long as no observation is made, but, according to the prevailing Copenhagen interpretation, the randomness caused by the wave function collapsing when an observation is made, is fundamental. This means that probability theory is required to describe nature. Others never came to terms with the loss of determinism. Albert Einstein famously remarked in a letter to Max Born: Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt. (I am convinced that God does not play dice). Although alternative viewpoints exist, such as that of quantum decoherence being the cause of an apparent random collapse, at present there is a firm consensus among physicists that probability theory is necessary to describe quantum phenomena.[How to reference and link to summary or text]
- Main article: Outline of probability
- The Logic of Statistical Inference, Ian Hacking, 1965
- The Emergence of Probability: A Philosophical Study of Early Ideas about Probability, Induction and Statistical Inference, Ian Hacking, Cambridge University Press, 2006, ISBN 0521685575, 9780521685573
- The Cambridge History of Seventeenth-century Philosophy, Daniel Garber, 2003
- Jeffrey, R.C., Probability and the Art of Judgment, Cambridge University Press. (1992). pp. 54-55 . ISBN 0-521-39459-7
- Franklin, J., The Science of Conjecture: Evidence and Probability Before Pascal, Johns Hopkins University Press. (2001). pp. 22, 113, 127
- Olofsson, Peter. (2005) Page 8.
- Olofsson, page 9
- Olofsson, page 35.
- Olofsson, page 29.
- Kallenberg, O. (2005) Probabilistic Symmetries and Invariance Principles. Springer -Verlag, New York. 510 pp. ISBN 0-387-25115-4
- Kallenberg, O. (2002) Foundations of Modern Probability, 2nd ed. Springer Series in Statistics. 650 pp. ISBN 0-387-95313-2
- Olofsson, Peter (2005) Probability, Statistics, and Stochastic Processes, Wiley-Interscience. 504 pp ISBN 0-471-67969-0.
- Damon Runyon, "It may be that the race is not always to the swift, nor the battle to the strong - but that is the way to bet."
- Pierre-Simon Laplace "It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge." Théorie Analytique des Probabilités, 1812.
- Richard von Mises "The unlimited extension of the validity of the exact sciences was a characteristic feature of the exaggerated rationalism of the eighteenth century" (in reference to Laplace). Probability, Statistics, and Truth, p 9. Dover edition, 1981 (republication of second English edition, 1957).
- Probability and Statistics EBook
- Edwin Thompson Jaynes. Probability Theory: The Logic of Science. Preprint: Washington University, (1996). — HTML index with links to PostScript files and PDF (first three chapters)
- People 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
- A tutorial on probability and Bayes’ theorem devised for first-year Oxford University students
- pdf file of An Anthology of Chance Operations (1963) at UbuWeb
- Probability Theory Guide for Non-Mathematicians
- Understanding Risk and Probability with BBC raw
Major fields of mathematics
Survival function - Kaplan-Meier - Logrank test - Failure rate - Proportional hazards models
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