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Probability is the likelihood or chance that something is the case or will happen. A related article is titled Uncertainty. For statistical certainty see Probability. Nihilism (from the Latin nihil, nothing is a philosophical position that argues that Existence is without objective meaning Purpose Agnosticism ( Greek: α- a-, without + γνώσις gnōsis, knowledge after Gnosticism) is the philosophical view that the Uncertainty is a term used in subtly different ways in a number of fields including Philosophy, Statistics, Economics, Finance, Insurance An approximation (represented by the symbol ≈ is an inexact representation of something that is still close enough to be useful Belief is the psychological state in which an individual holds a Proposition or Premise to be true Epistemology (from Greek επιστήμη - episteme, "knowledge" + λόγος, " Logos " or theory of knowledge A related article is titled Uncertainty. For statistical certainty see Probability. Determinism is the philosophical Proposition that every event including human cognition and behaviour decision and action is causally determined Probability theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. Probability theory is the branch of Mathematics concerned with analysis of random phenomena Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language

Contents

Interpretations

Main article: Probability interpretations

The word probability does not have a consistent direct definition. See also Philosophy of probability The word Probability has been used in a variety of ways since it was first coined in relation to Games of chance In fact, there are two broad categories of probability interpretations:

  1. Frequentists talk about probabilities only when dealing with well defined random experiments. Frequency probability is the interpretation of probability that defines an event's Probability as the limit of its relative frequency in a large Randomness is a lack of order Purpose, cause, or predictability The probability of a random event denotes the relative frequency of occurrence of an experiment's outcome, when repeating the experiment.
  2. Bayesians, however, assign probabilities to any statement whatsoever, even when no random process is involved. Bayesian probability interprets the concept of Probability as 'a measure of a state of knowledge'. Probability, for a Bayesian, is a way to represent an individual's degree of belief in a statement, given the evidence.

History

Further information: Statistics

The scientific study of probability is a modern development. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. 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. "[1]

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). Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the Blaise Pascal (blɛz paskal (June 19 1623 &ndash August 19 1662 was a French Mathematician, Physicist, and religious Philosopher Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject. Christiaan Huygens (ˈhaɪgənz in English ˈhœyɣəns in Dutch) ( April 14, 1629 &ndash July 8, 1695) was a Dutch Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics. For other family members named Jacob see Bernoulli family. Jacob Bernoulli (also known as James or Jacques) ( Basel Ars Conjectandi ( Latin: The Art of Conjecturing is a mathematical paper written by Jakob Bernoulli and published eight years after his death "Moivre" redirects here for the French commune see Moivre Marne. The Doctrine of Chances was the first textbook on Probability theory, written by 18th-century French Mathematician Abraham de Moivre and See Ian Hacking's The Emergence of Probability for a history of the early development of the very concept of mathematical probability. Ian Hacking, CC, PhD, FRSC, FBA (born February 18, 1936 in Vancouver) is a Canadian university

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. Roger Cotes FRS ( July 10, 1682 – June 5, 1716) was an English Mathematician, known for working closely with Thomas Simpson ( August 20, 1710 &ndash May 14, 1761) was a British Mathematician, Inventor and Eponym 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 y = φ(x), x being any error and y its probability, and laid down three properties of this curve:

  1. it is symmetric as to the y-axis;
  2. the x-axis is an asymptote, the probability of the error \infty being 0;
  3. the area enclosed is 1, it being certain that an error exists. An asymptote of a real-valued function y=f(x is a curve which describes the behavior of f as either x or y goes to infinity

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. Daniel Bernoulli ( Groningen, 29 January 1700 &ndash 27 July 1782 was a Dutch - Swiss Mathematician, who is particularly remembered for his applications

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). The method of least squares is used to solve Overdetermined systems Least squares is often applied in statistical contexts particularly Regression analysis. Adrien-Marie Legendre ( September 18 1752 – January 10 1833) was a French Mathematician. 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,

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

h being a constant depending on precision of observation, and c a scale factor ensuring that the area under the curve equals 1. Robert Adrain ( September 30, 1775 - August 10, 1843) was a Scientist and Mathematician, considered one of the most He gave two proofs, the second being essentially the same as John Herschel's (1850). Sir John Frederick William Herschel 1st Baronet KH, FRS ( March 7, 1792 &ndash May 11, 1871)was an Gauss gave the first proof which seems to have been known in Europe (the third after Adrain's) in 1809. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Further proofs were given by Laplace (1810, 1812), Gauss (1823), James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), W. Sir James Ivory ( February 17, 1765 &ndash September 21 1842) was a Scottish Mathematician. Friedrich Wilhelm Bessel (22 July 1784 &ndash 17 March 1846 was a German Mathematician, Astronomer, and systematizer of the Bessel functions F. Donkin (1844, 1856), and Morgan Crofton (1870). Morgan Crofton (born 1826 in Dublin, Ireland, died in 1915 in Brighton, England) was a mathematician who contributed to the field of geometric Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Augustus De Morgan ( 27 June, 1806 &ndash 18 March, 1871) was a British Mathematician and Logician. James Whitbread Lee Glaisher ( 5 November 1848 - 7 December 1928) son of James Glaisher, the meteorologist was a prolific English Honors and Awards Awards Gold Medal of the Royal Astronomical Society (1872 Bruce Medal (1902 Named Peters's (1856) formula for r, 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. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar Sylvestre François de Lacroix ( April 28, 1765 &ndash May 24, 1843) was a French Mathematician. Lambert Adolphe Jacques Quételet ( 22 February 1796 &ndash 17 February 1874) was a Belgian Astronomer, Mathematician Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important Paul Matthieu Hermann Laurent ( 2 September 1841 Echternach, Luxembourg - 19 February 1908 Paris, France Karl Pearson FRS ( March 27 1857 &ndash April 27 1936) established the disciplineof Mathematical statistics. Augustus De Morgan and George Boole improved the exposition of the theory. Augustus De Morgan ( 27 June, 1806 &ndash 18 March, 1871) was a British Mathematician and Logician. George Boole (buːl ( November 2, 1815 &ndash 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 is used in two ways which although related imply different views of the content of the subject

Mathematical treatment

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). In Probability theory, an event is a set of outcomes (a Subset of the Sample space) to which a probability is assigned 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". In Probability theory, one says that an event happens almost surely (a

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) = {1} - \tfrac{1}{6} = \tfrac{5}{6}. See Complementary event for a more complete treatment. In Probability theory, the complement of any event A is the event, i

If two events, A and B are independent then the joint probability is

P(A \mbox{ and }B) = P(A \cap B) = P(A) P(B),\,

for example if two coins are flipped the chance of both being heads is \tfrac{1}{2}\times\tfrac{1}{2} = \tfrac{1}{4}. In Probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other In the study of Probability, given two Random variables X and Y, the joint distribution of X and Y is the distribution

If two events are mutually exclusive then the probability of either occurring is

P(A\mbox{ or }B) = P(A \cup B)= P(A) + P(B).

For example, the chance of rolling a 1 or 2 on a six-sided die is P(1\mbox{ or }2) = P(1) + P(2) = \tfrac{1}{6} + \tfrac{1}{6} = \tfrac{1}{3}. In simple terms two events are mutually exclusive if they cannot occur at the same time (i

If the events are not mutually exclusive then

\mathrm{P}\left(A \hbox{ or } B\right)=\mathrm{P}\left(A\right)+\mathrm{P}\left(B\right)-\mathrm{P}\left(A \mbox{ and } B\right).

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 \tfrac{13}{52} + \tfrac{12}{52} - \tfrac{3}{52} = \tfrac{11}{26}, 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 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

P(A \mid B) = \frac{P(A \cap B)}{P(B)}.\,

If P(B) = 0 then P(A \mid B) is undefined. In Mathematics, defined and undefined are used to explain whether or not expressions have meaningful sensible and unambiguous values

Summary of probabilities
Event Probability
A P(A)\in[0,1]\,
not A P(A')=1-P(A)\,
A or B \begin{align} P(A\cup B) & = P(A)+P(B)-P(A\cap B) \\ & = P(A)+P(B) \qquad\mbox{if A and B are mutually exclusive}\\ \end{align}
A and B \begin{align} P(A\cap B) & = P(A|B)P(B) \\ & = P(A)P(B) \qquad\mbox{if A and B are independent}\\ \end{align}
A given B P(A|B)\,

Theory

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. Probability theory is the branch of Mathematics concerned with analysis of random phenomena The word theory has many distinct meanings in different fields of Knowledge, depending on their methodologies and the context of discussion. Probability theory is the branch of Mathematics concerned with analysis of random phenomena 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. Andrey Nikolaevich Kolmogorov (Андрей Николаевич Колмогоров ( April 25, 1903 - October 20, 1987) was a Soviet Richard Threlkeld Cox ( 1898 - May 2, 1991) was a professor of Physics at Johns Hopkins University, known for Cox's theorem In Kolmogorov's formulation (see probability space), sets are interpreted as events and probability itself as a measure on a class of sets. A probability space, in Probability theory, is the conventional Mathematical model of Randomness. In Probability theory, an event is a set of outcomes (a Subset of the Sample space) to which a probability is assigned In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with 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. Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of Probability theory from a certain set of postulates In both cases, the laws of probability are the same, except for technical details. In Probability theory, the Probability P of some event E, denoted P(E is defined in such a way that P satisfies the

There are other methods 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. The Dempster-Shafer theory is a mathematical theory of Evidence based on belief functions and plausible reasoning, which is used to combine separate pieces Possibility theory is a mathematical theory for dealing with certain types of Uncertainty and is an alternative to Probability theory.

Applications

Two major applications of probability theory in everyday life are in risk assessment and in trade on commodity markets. Risk is a Concept that denotes the precise probability of specific eventualities Commodity markets are markets where raw or primary products are exchanged 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. Environmental law is a complex and interlocking body of Statutes, Common law, Treaties, conventions Regulations and policies which very Quality of life is the degree of well-being felt by an individual or group of people 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. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Risk is a Concept that denotes the precise probability of specific eventualities 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. The law of small numbers may refer to ''The Law of Small Numbers'' (book, authored by Ladislaus Bortkiewicz The

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. Behavioral economics and behavioral finance are closely related fields which apply scientific research on human and social cognitive and emotional factors to better Groupthink is a type of thought exhibited by group members who try to minimize conflict and reach consensus without critically testing analyzing and evaluating ideas

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. Democracy is a form of government in which the supreme power is held completely by the people under a free electoral system

Another significant application of probability theory in everyday life is reliability. Reliability theory of aging and longevity is a scientific approach aimed to gain theoretical insights into mechanisms of biological Aging and species survival patterns by applying 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. Reliability theory developed apart from the mainstream of Probability and Statistics. The probability of failure is also closely associated with the product's warranty. In commercial and consumer transactions a warranty is an Obligation or guarantee that an article or service sold is as factually stated or legally

Relation to randomness

Main article: Randomness

In a deterministic universe, based on Newtonian concepts, there is no probability if all conditions are known. Randomness is a lack of order Purpose, cause, or predictability Determinism is the philosophical Proposition that every event including human cognition and behaviour decision and action is causally determined Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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 analysing 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\cdot 10^{23}) that only statistical description of its properties is feasible. Kinetic theory (or kinetic theory of gases) attempts to explain Macroscopic properties of Gases such as pressure temperature or volume by considering The Avogadro constant (symbols L, N A also called Avogadro's number, is the number of "elementary entities" (usually Atoms

A revolutionary discovery of 20th century physics was the random character of all physical processes that occur at microscopic scales and are governed by the laws of quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons 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. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system The Copenhagen interpretation is an interpretation of Quantum mechanics. In certain interpretations of quantum mechanics, wave function collapse is one of two processes by which Quantum systems apparently evolve according to the laws of This means that probability theory is required to describe nature. Probability theory is the branch of Mathematics concerned with analysis of random phenomena 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. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical Max Born (11 December 1882 &ndash 5 January 1970 was a German Physicist and Mathematician who was instrumental in the development of Quantum (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 the physicists that probability theory is necessary to describe quantum phenomena. In Quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior

See also

Footnotes

  1. ^ Jeffrey, R. Decision theory in Mathematics and Statistics is concerned with identifying the Values uncertainties and other issues relevant in a given Equiprobability is a philosophical concept in Probability theory that allows one to assign equal probabilities to outcomes that are judged to be Equipossible Fuzzy measure theory considers a number of special classes of measures each of which is characterized by a special property Game theory is a branch of Applied mathematics that is used in the Social sciences (most notably Economics) Biology, Engineering, Information theory is a branch of Applied mathematics and Electrical engineering involving the quantification of Information. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with 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 capacity of Probability theory to handle uncertainty with the capacity of Deductive A random field is a generalization of a stochastic process such that the underlying parameter need no longer be a simple real but can instead be a multidimensional vector space or even a manifold A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Please add any Wikipedia articles related to Statistics that are not already on this list A stochastic process, or sometimes random process, is the counterpart to a deterministic process (or Deterministic system) in Probability theory. In Mathematics, the Wiener process is a continuous-time Stochastic process named in honor of Norbert Wiener. For Taleb's book on the subject see The Black Swan. The black swan theory refers to a large-impact hard-to-predict and rare event beyond the C. , Probability and the Art of Judgment, Cambridge University Press. (1992). pp. 54-55 . ISBN 0-521-39459-7

Sources

Quotations

External links

Dictionary

probability

-noun

  1. the state of being probable; likelihood
  2. an event that is likely to occur
  3. the relative likelihood of an event happening
  4. (mathematics) a number, between 0 and 1, expressing the precise likelihood of an event happening
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