Informally, probable is one of several words applied to uncertain events or knowledge, being closely related in meaning to likely, risky, hazardous, and doubtful. Chance, odds, and bet are other words expressing similar notions. Just as the theory of mechanics assigns precise definitions to such everyday terms as work and force, the theory of probability attempts to quantify the notion of probable.Probability always lies between 0 and 1. If probability is equal to 1 then that event is certain to happen and if the probability is 0 then that event will never occur. Nihilism is a philosophical position, often associated with Friedrich Nietzsche, which argues that the world, and especially past and current human existence, is without objective meaning, purpose, comprehensible truth, or essential value. ... Agnosticism is the philosophical view that the (truth) values of certain claims—particularly theological claims regarding the existence of God, gods, or deities—are unknown, inherently unknowable, or incoherent, and therefore, (some agnostics may go as far to say) irrelevant to life. ... // Relation between uncertainty, probability and risk In his seminal work Risk, Uncertainty, and Profit, Frank Knight (1921) established the important distinction between risk and uncertainty: … Uncertainty must be taken in a sense radically distinct from the familiar notion of Risk, from which it has never been properly separated. ... It has been suggested that this article or section be merged with estimation. ... Wiktionary has related dictionary definitions, such as: belief Belief is usually defined as a conviction to the truth of a proposition. ... Epistemology or theory of knowledge is the branch of philosophy that studies the nature and scope of knowledge. ... Certainty is the absence of all doubt. ... Determinism is the philosophical proposition that every event, including human cognition and action, is causally determined by an unbroken chain of prior occurrences. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Probability theory is the mathematical study of probability. ...

Historical remarks

Further information: 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. A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Gambling has had many different meanings depending on the cultural and historical context in which it is used. ...

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 for a history of the early development of the very concept of mathematical probability. Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parlement of Toulouse, southwestern France, and a mathematician who is given credit for his contribution towards the development of modern calculus. ... Blaise Pascal (pronounced []), (June 19, 1623 – August 19, 1662) was a French mathematician, physicist, and religious philosopher. ... Christiaan Huygens Christiaan Huygens (pronounced in English (IPA): ; in Dutch: )(April 14, 1629–July 8, 1695), was a Dutch mathematician and physicist; born in The Hague as the son of Constantijn Huygens. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Abraham de Moivre. ... The Doctrine of Chances is a book on probability theory by 18th-century French mathematician Abraham de Moivre, published in 1733. ... Ian Hacking, CC (born 1936) in Vancouver is a philosopher operating in the fields of philosophy of science and philosophy of language. ...

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. Roger Cotes (Burbage, Leicestershire July 10, 1682 – June 5, 1716 in Cambridge, Cambridgeshire) was an English mathematician. ... Thomas Simpson (August 20, 1110 – May 14, 1761) was a British mathematician, inventor and eponym of Simpsons rule to approximate definite integrals. ...

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 y = φ(x), x being any error and y its probability, and laid down three properties of this curve: To meet Wikipedias quality standards, this article or section may require cleanup. ...

1. it is symmetric as to the y-axis;
2. the x-axis is an asymptote, the probability of the error being 0;
3. the area enclosed is 1, it being certain that an error exists.

He deduced a formula for the mean of three observations. 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. An asymptote is a straight line or curve which a curve approaches as one moves along the curve. ... Daniel Bernoulli Daniel Bernoulli (Groningen, February 8, 1700 – Basel, March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland. ...

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, Least squares is a mathematical optimization technique that attempts to find a best fit to a set of data by attempting to minimize the sum of the squares of the differences (called residuals) between the fitted function and the data. ... Adrien-Marie Legendre (September 18, 1752 - January 10, 1833) was a French mathematician. ... Robert Adrain (September 30, 1775 - August 10, 1843) was a scientist and mathematician. ...

c and h being constants depending on precision of observation. 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 r, the probable error of a single observation, is well known. John Herschel Sir John Frederick William Herschel (7 March 1792 – 11 May 1871) was an English mathematician and astronomer. ... (30 April 1777 – 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... Sir James Ivory (1765 - September 21, 1842) was a Scottish mathematician. ... Friedrich Wilhelm Bessel (July 22, 1784 – March 17, 1846) was a German mathematician, astronomer, and systematizer of the Bessel functions (which, despite their name, were discovered by Daniel Bernoulli). ... Morgan Crofton (born 1826 in Dublin, Ireland, died in 1915 in Brighton, England) was a mathematician who contributed to the field of geometric probability theory. ... Augustus De Morgan (June 27, 1806 – March 18, 1871) was an Indian-born British mathematician and logician. ... Glaisher may mean James Glaisher the meteorologist; James Whitbread Lee Glaisher, the mathematician. ... Giovanni Virginio Schiaparelli (March 14, 1835 – July 4, 1910) was an Italian astronomer. ...

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. Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... Pierre-Simon Laplace Pierre-Simon Laplace (March 23, 1749 – March 5, 1827) was a French mathematician and astronomer, the discoverer of the Laplace transform and Laplaces equation. ... Sylvestre François de Lacroix (April 28, 1765–May 24, 1843) was a French mathematician. ... Lambert Adolphe Jacques Quételet (February 22, 1796 - 1874) was a Belgian astronomer, mathematician, statistician and sociologist. ... Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ... Paul Matthieu Hermann Laurent (2 September 1841 Echternach, Luxembourg - 19 February 1908 Paris, France) was a French mathematician. ... Karl Pearson (pencil sketch in notebook; there is some see-through of writing on next page) Karl Pearson (March 27, 1857 – April 27, 1936) was a major contributor to the early development of statistics as a serious scientific discipline in its own right. ... Augustus De Morgan (June 27, 1806 – March 18, 1871) was an Indian-born British mathematician and logician. ... George Boole [], (November 2, 1815 – December 8, 1864) was a British mathematician and philosopher. ...

On the geometric side (see integral geometry) contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin). In mathematics, the term integral geometry in is used in two ways, which, although related, imply different views of the content of the subject. ...

Concepts

There is essentially one set of mathematical rules for manipulating probability; these rules are listed under "Formalization of probability" below. (There are other rules for quantifying uncertainty, such as the Dempster-Shafer theory and possibility theory, but those are essentially different and not compatible with the laws of probability as they are usually understood.) However, there is ongoing debate over what, exactly, the rules apply to; this is the topic of probability interpretations. The Dempster-Shafer theory is a mathematical theory of evidence [SH76] based on belief functions and plausible reasoning, which is used to combine separate pieces of information (evidence) to calculate the probability of an event. ... Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. ... The word probability has been used in a variety of ways since it was first coined in relation to games of chance. ...

The general idea of probability is often divided into two related concepts:

• Aleatory probability, which represents the likelihood of future events whose occurrence is governed by some random physical phenomenon. This concept can be further divided into physical phenomena that are predictable, in principle, with sufficient information (see Determinism), and phenomena which are essentially unpredictable. Examples of the first kind include tossing dice or spinning a roulette wheel; an example of the second kind is radioactive decay.

• Epistemic probability, which represents one's uncertainty about propositions when one lacks complete knowledge of causative circumstances. Such propositions may be about past or future events, but need not be. Some examples of epistemic probability are to assign a probability to the proposition that a proposed law of physics is true, and to determine how "probable" it is that a suspect committed a crime, based on the evidence presented.

It is an open question whether aleatory probability is reducible to epistemic probability based on our inability to precisely predict every force that might affect the roll of a die, or whether such uncertainties exist in the nature of reality itself, particularly in quantum phenomena governed by Heisenberg's uncertainty principle. Although the same mathematical rules apply regardless of which interpretation is chosen, the choice has major implications for the way in which probability is used to model the real world. Statistical regularity has motivated the development of the relative frequency concept of probability. ... The word random is used to express lack of purpose, cause, order, or predictability in non-scientific parlance. ... Determinism is the philosophical proposition that every event, including human cognition and action, is causally determined by an unbroken chain of prior occurrences. ... Typical role-playing dice, showing a variety of colors and styles. ... Roulette is a casino and gambling game (Roulette is a French word meaning small wheel). A croupier turns a round roulette wheel which has 37 or 38 separately numbered pockets in which a ball must land. ... Radioactive decay is the set of various processes by which unstable atomic nuclei emit subatomic particles (radiation). ... Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of statements, or to the degree of belief of rational agents in the truth of statements; when used with Bayes theorem, it then becomes Bayesian inference. ... Fig. ... In quantum physics, the Heisenberg uncertainty principle or the Heisenberg indeterminacy principle — the latter name given to it by Niels Bohr — states that one cannot measure values (with arbitrary precision) of certain conjugate quantities, which are pairs of observables of a single elementary particle. ...

Formalization of probability

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, sets are interpreted as events and probability itself as a measure on a class of sets. In Cox's formulation, 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; The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ... Probability theory is the mathematical study of probability. ... Andrey Kolmogorov Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a... Richard Threlkeld Cox (1898 - May 2, 1991) was a professor of physics at Johns Hopkins University, known for Coxs theorem relating to the foundations of probability. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. ... In mathematics, a measure is a function that assigns a number, e. ...

1. a probability is a number between 0 and 1;
2. the probability of an event or proposition and its complement must add up to 1; and
3. the joint probability of two events or propositions is the product of the probability of one of them and the probability of the second, conditional on the first.

The reader will find an exposition of the Kolmogorov formulation in the probability theory article, and of the Cox formulation in the Cox's theorem article. See also the article on probability axioms. This article defines some terms which characterize probability distributions of two or more variables. ... This article defines some terms which characterize probability distributions of two or more variables. ... Probability theory is the mathematical study of probability. ... Coxs theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. ... The probability P of some event E, denoted , is defined with respect to a universe, or sample space , of all possible elementary events in such a way that P must satisfy the Kolmogorov axioms. ...

For an algebraic alternative to Kolmogorov's approach, see algebra of random variables. In the algebraic axiomatization of probability theory, one of whose main proponents was Irving Segal, the primary concept is not that of probability of an event, but rather that of a random variable. ...

Representation and interpretation of probability values

The probability of an event is generally represented as a real number between 0 and 1, inclusive. An impossible event has a probability of exactly 0, and a certain event has a probability of 1, but 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". In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ... In mathematics, specifically, in probability theory, the phrase almost surely is a concise, precise way to state except on a set or event of probability measure zero. ...

Most probabilities that occur in practice are numbers between 0 and 1, indicating the event's position on the continuum between impossibility and certainty. The closer an event's probability is to 1, the more likely it is to occur.

For example, if two mutually exclusive events are assumed equally probable, such as a flipped or spun coin landing heads-up or tails-up, we can express the probability of each event as "1 in 2", or, equivalently, "50%" or "1/2". In logic, two mutually exclusive (or mutual exclusive according to some sources) propositions are propositions that logically cannot both be true. ...

Probabilities are equivalently expressed as odds, which is the ratio of the probability of one event to the probability of all other events. The odds of heads-up, for the tossed/spun coin, are (1/2)/(1 - 1/2), which is equal to 1/1. This is expressed as "1 to 1 odds" and often written "1:1". In probability theory and statistics the odds in favor of an event or a proposition are the quantity p / (1 − p), where p is the probability of the event or proposition. ...

Odds a:b for some event are equivalent to probability a/(a+b). For example, 1:1 odds are equivalent to probability 1/2, and 3:2 odds are equivalent to probability 3/5.

There remains the question of exactly what can be assigned probability, and how the numbers so assigned can be used; this is the question of probability interpretations. There are some who claim that probability can be assigned to any kind of an uncertain logical proposition; this is the Bayesian interpretation. There are others who argue that probability is properly applied only to random events as outcomes of some specified random experiment, for example sampling from a population; this is the frequentist interpretation. There are several other interpretations which are variations on one or the other of those, or which have less acceptance at present. The word probability has been used in a variety of ways since it was first coined in relation to games of chance. ... In the philosophy of mathematics Bayesianism is the tenet that the mathematical theory of probability is applicable to the degree to which a person believes a proposition. ... Statistical regularity has motivated the development of the relative frequency concept of probability. ...

Distributions

A probability distribution is a function that assigns probabilities to events or propositions. For any set of events or propositions there are many ways to assign probabilities, so the choice of one distribution or another is equivalent to making different assumptions about the events or propositions in question. In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...

There are several equivalent ways to specify a probability distribution. Perhaps the most common is to specify a probability density function. Then the probability of an event or proposition is obtained by integrating the density function. The distribution function may also be specified directly. In one dimension, the distribution function is called the cumulative distribution function. Probability distributions can also be specified via moments or the characteristic function, or in still other ways. In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In calculus, the integral of a function is a generalization of area, mass, volume and total. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... -1... In probability theory, the characteristic function of any random variable completely defines its probability distribution. ...

A distribution is called a discrete distribution if it is defined on a countable, discrete set, such as a subset of the integers. A distribution is called a continuous distribution if it has a continuous distribution function, such as a polynomial or exponential function. Most distributions of practical importance are either discrete or continuous, but there are examples of distributions which are neither. In mathematics the term countable set is used to describe the size of a set, e. ... Look up discrete in Wiktionary, the free dictionary. ...

Important discrete distributions include the discrete uniform distribution, the Poisson distribution, the binomial distribution, the negative binomial distribution, and the Maxwell-Boltzmann distribution. In mathematics, the uniform distributions are simple probability distributions. ... In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ... In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. ... In probability and statistics the negative binomial distribution is a discrete probability distribution. ... The introduction to this article provides insufficient context for those unfamiliar with the subject matter. ...

Important continuous distributions include the normal distribution, the gamma distribution, the Student's t-distribution, and the exponential distribution. The normal distribution, also called Gaussian distribution (although Gauss was not the first to work with it), is an extremely important probability distribution in many fields. ... In probability theory and statistics, the gamma distribution is a continuous probability distribution. ... In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ... In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ...

Probability in mathematics

Probability axioms form the basis for mathematical probability theory. Calculation of probabilities can often be determined using combinatorics or by applying the axioms directly. Probability applications include even more than statistics, which is usually based on the idea of probability distributions and the central limit theorem. The probability P of some event E, denoted , is defined with respect to a universe, or sample space , of all possible elementary events in such a way that P must satisfy the Kolmogorov axioms. ... Probability theory is the mathematical study of probability. ... Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... A central limit theorem is any of a set of weak-convergence results in probability theory. ...

To give a mathematical meaning to probability, consider flipping a "fair" coin. Intuitively, the probability that heads will come up on any given coin toss is "obviously" 50%; but this statement alone lacks mathematical rigor. Certainly, while we might expect that flipping such a coin 10 times will yield 5 heads and 5 tails, there is no guarantee that this will occur; it is possible, for example, to flip 10 heads in a row. What then does the number "50%" mean in this context? ...

One approach is to use the law of large numbers. In this case, we assume that we can perform any number of coin flips, with each coin flip being independent—that is to say, the outcome of each coin flip is unaffected by previous coin flips. If we perform N trials (coin flips), and let N_H be the number of times the coin lands heads, then we can, for any N, consider the ratio . The law of large numbers is a fundamental concept in statistics and probability that describes how the average of a randomly selected sample from a large population is likely to be close to the average of the whole population. ...

As N gets larger and larger, we expect that in our example the ratio will get closer and closer to 1/2. This allows us to "define" the probability of flipping heads as the limit, as N approaches infinity, of this sequence of ratios: In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...

In actual practice, of course, we cannot flip a coin an infinite number of times; so in general, this formula most accurately applies to situations in which we have already assigned an a priori probability to a particular outcome (in this case, our assumption that the coin was a "fair" coin). The law of large numbers then says that, given Pr(H), and any arbitrarily small number ε, there exists some number n such that for all N > n,

In other words, by saying that "the probability of heads is 1/2", we mean that, if we flip our coin often enough, eventually the number of heads over the number of total flips will become arbitrarily close to 1/2; and will then stay at least as close to 1/2 for as long as we keep performing additional coin flips.

Note that a proper definition requires measure theory, which provides means to cancel out those cases where the above limit does not provide the "right" result (or is even undefined) by showing that those cases have a measure of zero. In mathematics, a measure is a function that assigns a number, e. ...

The a priori aspect of this approach to probability is sometimes troubling when applied to real world situations. For example, in the play Rosencrantz & Guildenstern Are Dead by Tom Stoppard, a character flips a coin which keeps coming up heads over and over again, a hundred times. He can't decide whether this is just a random event—after all, it is possible (although unlikely) that a fair coin would give this result—or whether his assumption that the coin is fair is at fault. DVD cover Rosencrantz & Guildenstern Are Dead is a humorous, absurdist, tragic and existentialist play by Tom Stoppard, first staged in 1966. ... Sir Tom Stoppard OM, CBE (born Tomáš Straussler on 3 July 1937) is a British playwright. ...

Remarks on probability calculations

The difficulty of probability calculations lies in setting up the problem in an appropriate way. (There is never a uniquely correct way to set up a problem, but some ways are better than others.) Especially difficult is drawing meaningful conclusions from the probabilities calculated. An amusing probability riddle, the Monty Hall problem, demonstrates the pitfalls nicely. In search of a new car, the player picks door 1. ...

To learn more about the basics of probability theory, see the article on probability axioms and the article on Bayes' theorem, which explains the use of conditional probabilities in cases where the occurrence of two events is related. Probability theory is the mathematical study of probability. ... The probability of some event (denoted ) is defined with respect to a universe or sample space of all possible elementary events in such a way that must satisfy the Kolmogorov axioms. ... Bayess theorem (also known as Bayess rule) is a result in probability theory, which relates the conditional and marginal probability distributions of random variables. ...

Applications of probability theory to everyday life

Two major applications of probability theory in everyday life are in risk assessment and in trade on commodity markets. Governments typically apply probability methods in environmental regulation where it is called "pathway analysis", and are 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. It is not correct to say that statistics are involved in the modelling itself, as typically the assessments of risk are one-time and thus require more fundamental probability models, e.g. "the probability of another 9/11". A law of small numbers tends to apply to all such choices and perception of the effect of such choices, which makes probability measures a political matter. Risk is a concept which relates to human expectations. ... This article or section is missing references or citation of sources. ... Environmental law is a body of law which addresses the system of complex and interlocking rules which seeks to protect from destruction or development certain species or favored natural areas thought to be endangered by human encroachment. ... The well-being or quality of life of a population is an important concern in economics and political science. ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Risk is a concept which relates to human expectations. ... The law of small numbers may refer to the specific features of the Poisson distribution, as in the book The Law of Small Numbers by Ladislaus Bortkiewicz; or the tendency for an initial segment of data to show some bias that drops out later (one example in number theory being...

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 trade 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. Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel winner Daniel Kahneman, was an important figure in the development of behavioral finance and economics and continues to write extensively in the field. ... It has been suggested that this article or section be merged with Pluralistic ignorance. ...

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. A good example is the application of game theory, itself based strictly on probability, to the Cold War and the mutual assured destruction doctrine. 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. Game theory is a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their returns. ... For other uses, please see Cold War (disambiguation). ... Mutual assured destruction (MAD) is a doctrine of military strategy in which a full-scale use of nuclear weapons by one of two opposing sides would effectively result in the destruction of both the attacker and the defender. ...

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 is also closely associated with the product's warranty. Reliability concerns quality or consistency. ... Car redirects here. ... Reliability theory developed apart from the mainstream of probability and statistics, and was used originally as a tool to help nineteenth century maritime insurance and life insurance companies compute profitable rates to charge their customers. ... In business law, a warranty is a promise that something sold is as factually stated or legally implied by the seller. ...

See also

• Bayesian probability
• Bernoulli process
• Cox's theorem
• Decision theory
• Fuzzy measure theory
• Frequency probability
• Game of chance
• Game theory
• Information theory
• Law of averages
• Law of large numbers
• Measure theory
• Expected value
• Normal distribution
• Probabilistic argumentation
• Probabilistic logic
• Probability theory
• Random fields
• Random variable
• Statistics
  • List of statistical topics
• Stochastic process
• Wiener process
• Important publications in probability
• p-value

In the philosophy of mathematics Bayesianism is the tenet that the mathematical theory of probability is applicable to the degree to which a person believes a proposition. ... In probability and statistics, a Bernoulli process is a discrete_time stochastic process consisting of finite or infinite sequence of independent random variables X1, X2, X3,..., such that For each i, the value of Xi is either 0 or 1; For all values of i, the probability that Xi = 1 is... Coxs theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. ... Decision theory is an interdisciplinary area of study, related to and of interest to practitioners in mathematics, statistics, economics, philosophy, management and psychology. ... Fuzzy measure theory considers a number of special classes of measures, each of which is characterized by a special property. ... Statistical regularity has motivated the development of the relative frequency concept of probability. ... A game of chance is a game whose outcome is strongly influenced by some randomizing device, and upon which contestants frequently wager money. ... Game theory is a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their returns. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... The law of averages is a lay term used to express the view that eventually, everything evens out. ... The law of large numbers is a fundamental concept in statistics and probability that describes how the average of a randomly selected sample from a large population is likely to be close to the average of the whole population. ... In mathematics, a measure is a function that assigns a number, e. ... In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are... The normal distribution, also called Gaussian distribution (although Gauss was not the first to work with it), is an extremely important probability distribution in many fields. ... Probabilistic argumentation is a general theory of reasoning under uncertainty and ignorance. ... The aim of a probabilistic logic (or probability logic) is to combine the capacitiy of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure. ... Probability theory is the mathematical study of probability. ... In probability theory, let S = {X1, ..., Xn}, with the Xi in {0,1,...,G-1}, be a set of random variables on the sample space Ω={0,1,...,G-1}n, a probability measure π is a random field if . There exist several types of random fields, such as Markov... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Please add any Wikipedia articles related to statistics that are not already on this list. ... In the mathematics of probability, a stochastic process is a random function. ... In mathematics, the Wiener process, so named in honor of Norbert Wiener, is a continuous-time Gaussian stochastic process with independent increments used in modelling Brownian motion and some random phenomena observed in finance. ... // Probability The Doctrine of Chances Author: Abraham de Moivre Publication data: 1738 (2nd ed. ... In statistical hypothesis testing, the p-value of a random variable T used as a test statistic is the probability that T will assume a value at least as extreme as the observed value tobserved, given that a null hypothesis being considered is true. ...

Literature

• Olav Kallenberg, Probabilistic Symmetries and Invariance Principles. Springer -Verlag, New York (2005). 510 pp. ISBN 0-387-25115-4
• Kallenberg, O., Foundations of Modern Probability, 2nd ed. Springer Series in Statistics. (2002). 650 pp. ISBN 0-387-95313-2

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Quotations

• 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).

Damon Runyon Damon Runyon (October 4, 1884 – December 10, 1946) was a newspaperman and writer. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Richard von Mises. ...

External links

• Probability in Wikibooks
• A Collection of articles on Probability, many of which are accompanied by Java simulations at cut-the-knot
• Edwin Thompson Jaynes. Probability Theory: The Logic of Science. Preprint: Washington University, (1996). -- HTML and PDF
• An online probability textbook which uses computer programming as a teaching aid
• "The Not So Random Coin Toss, Mathematicians Say Slight but Real Bias Toward Heads". NPR.
• Figuring the Odds (Probability Puzzles)
• Dictionary of the History of Ideas: Certainty in Seventeenth-Century Thought
• Dictionary of the History of Ideas: Certainty since the Seventeenth Century
• Small thought experiment on toss of a coin:
• The Earthquake Paradox: A Probability Puzzle -- from The Econophysics Blog
• Probability and induction
• Interpretations of Probability