Probability

Probability

Overview
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we are
not certain. The proposition of interest is usually of the form "Will a specific event
Event (probability theory)
In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...

 occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1, we call probability.
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Encyclopedia
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we are
not certain. The proposition of interest is usually of the form "Will a specific event
Event (probability theory)
In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...

 occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The certainty we adopt can be described in terms of a numerical measure and this number, between 0 and 1, we call probability. The higher the probability of an event, the more certain we are that the event will occur. Thus, probability in an applied sense is a measure of the likeliness that a (random) event will occur.

The concept has been given an axiomatic mathematical
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 derivation in probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, which is used widely in such areas of study as mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, finance
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...

, gambling
Gambling
Gambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods...

, science
Science
Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

, artificial intelligence
Artificial intelligence
Artificial intelligence is the intelligence of machines and the branch of computer science that aims to create it. AI textbooks define the field as "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions that maximize its...

/machine learning
Machine learning
Machine learning, a branch of artificial intelligence, is a scientific discipline concerned with the design and development of algorithms that allow computers to evolve behaviors based on empirical data, such as from sensor data or databases...

 and philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...

 to, for example, draw inferences about the likeliness of events. Probability is used to describe the underlying mechanics and regularities of complex systems
Complex systems
Complex systems present problems in mathematical modelling.The equations from which complex system models are developed generally derive from statistical physics, information theory and non-linear dynamics, and represent organized but unpredictable behaviors of systems of nature that are considered...

.

Interpretations


The word probability does not have a singular direct definition
Definition
A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...

 for practical application. In fact, there are several broad categories of probability interpretations, whose adherents possess different (and sometimes conflicting) views about the fundamental nature of probability. For example:
  1. Frequentists talk about probabilities only when dealing with experiments that are random and well-defined
    Well-defined
    In mathematics, well-definition is a mathematical or logical definition of a certain concept or object which uses a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy...

    . 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.
  2. Subjectivists assign numbers per subjective probability, i.e., as a degree of belief.
  3. Bayesians
    Bayesian probability
    Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...

     include expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by a prior probability distribution. The data is incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a posterior probability distribution that incorporates all the information known to date.

Etymology


The word Probability derives
Derivation (linguistics)
In linguistics, derivation is the process of forming a new word on the basis of an existing word, e.g. happi-ness and un-happy from happy, or determination from determine...

 from the Latin probabilitas, which can also mean probity, a measure of the authority
Authority
The word Authority is derived mainly from the Latin word auctoritas, meaning invention, advice, opinion, influence, or command. In English, the word 'authority' can be used to mean power given by the state or by academic knowledge of an area .-Authority in Philosophy:In...

 of a witness
Witness
A witness is someone who has firsthand knowledge about an event, or in the criminal justice systems usually a crime, through his or her senses and can help certify important considerations about the crime or event. A witness who has seen the event first hand is known as an eyewitness...

 in a legal case
Legal case
A legal case is a dispute between opposing parties resolved by a court, or by some equivalent legal process. A legal case may be either civil or criminal...

 in Europe
Europe
Europe is, by convention, one of the world's seven continents. Comprising the westernmost peninsula of Eurasia, Europe is generally 'divided' from Asia to its east by the watershed divides of the Ural and Caucasus Mountains, the Ural River, the Caspian and Black Seas, and the waterways connecting...

, and often correlated with the witness's nobility
Nobility
Nobility is a social class which possesses more acknowledged privileges or eminence than members of most other classes in a society, membership therein typically being hereditary. The privileges associated with nobility may constitute substantial advantages over or relative to non-nobles, or may be...

. In a sense, this differs much from the modern meaning of dhinki chika, which, in contrast, is a measure of the weight of empirical evidence, and is arrived at from inductive reasoning
Inductive reasoning
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...

 and statistical inference
Statistical inference
In statistics, statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation...

.

History


The scientific study of probability is a modern development. Gambling
Gambling
Gambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods...

 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 of course, 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.

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 elementary work by Girolamo Cardano in the 16th century, the doctrine of probabilities dates to the correspondence of Pierre de Fermat
Pierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...

 and Blaise Pascal
Blaise Pascal
Blaise Pascal , was a French mathematician, physicist, inventor, writer and Catholic philosopher. He was a child prodigy who was educated by his father, a tax collector in Rouen...

 (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject. Jakob Bernoulli's Ars Conjectandi
Ars Conjectandi
Ars Conjectandi is a combinatorial mathematical paper written by Jakob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, most notably among other combinatorial topics, probability theory: indeed, it is widely regarded as...

(posthumous, 1713) and Abraham de Moivre's
Abraham de Moivre
Abraham de Moivre was a French mathematician famous for de Moivre's formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmund Halley, and James Stirling...

 Doctrine of Chances
Doctrine of chances
In law, the doctrine of chances is a rule of evidence that allows evidence to show that it is unlikely a defendant would be repeatedly, innocently involved in similar, suspicious circumstances....

(1718) treated the subject as a branch of mathematics. See Ian Hacking's
Ian Hacking
Ian Hacking, CC, FRSC, FBA is a Canadian philosopher, specializing in the philosophy of science.- Life and works :...

 The Emergence of Probability and James Franklin's
James Franklin (philosopher)
James Franklin is an Australian philosopher, mathematician and historian of ideas. He was educated at St. Joseph's College, Hunters Hill, New South Wales. His undergraduate work was at the University of Sydney , where he attended St John's College and he was influenced by philosophers David Stove...

 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
Roger Cotes
Roger Cotes FRS was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the Principia, before publication. He also invented the quadrature formulas known as Newton–Cotes formulas and first introduced what is known today as...

 Opera Miscellanea (posthumous, 1722), but a memoir prepared by Thomas Simpson
Thomas Simpson
Thomas Simpson FRS was a British mathematician, inventor and eponym of Simpson's rule to approximate definite integrals...

 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 certain assignable limits define the range of all errors. Simpson also discusses continuous errors and describes a probability curve.

Pierre-Simon Laplace
Pierre-Simon Laplace
Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...

 (1774) first tried to deduce a rule for combining 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:
  1. It is symmetric as to the -axis;
  2. The -axis is an asymptote
    Asymptote
    In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...

    , the probability of the error being 0;
  3. The area enclosed is 1, it being certain that an error exists.

He also provided, in 1781, a formula for the law of facility of error (a term Lagrange used in 1774), but it led to unmanageable equations. Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

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

Adrien-Marie Legendre
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...

 (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). In ignorance of Legendre's contribution, an Irish-American writer, Robert Adrain
Robert Adrain
Robert Adrain was a scientist and mathematician, considered one of the most brilliant mathematical minds of the time in America....

, 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
John Herschel
Sir John Frederick William Herschel, 1st Baronet KH, FRS ,was an English mathematician, astronomer, chemist, and experimental photographer/inventor, who in some years also did valuable botanical work...

 (1850). Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

 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
James Ivory (mathematician)
Sir James Ivory was a Scottish mathematician.Ivory was born in Dundee and attended Dundee Grammar School. In 1779 he entered the University of St Andrews, distinguishing himself especially in mathematics...

 (1825, 1826), Hagen (1837), Friedrich Bessel
Friedrich Bessel
-References:* John Frederick William Herschel, A brief notice of the life, researches, and discoveries of Friedrich Wilhelm Bessel, London: Barclay, 1847 -External links:...

 (1838), W. F. Donkin (1844, 1856), and Morgan Crofton
Morgan Crofton
Morgan Crofton was a mathematician who contributed to the field of geometric probability theory. He also worked with James Joseph Sylvester and contributed an article on probability to the 9th edition of the Encyclopædia Britannica...

 (1870). Other contributors were Ellis (1844), De Morgan
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....

 (1864), Glaisher
James Whitbread Lee Glaisher
James Whitbread Lee Glaisher son of James Glaisher, the meteorologist, was a prolific English mathematician.He was educated at St Paul's School and Trinity College, Cambridge, where he was second wrangler in 1871...

 (1872), and Giovanni Schiaparelli
Giovanni Schiaparelli
Giovanni Virginio Schiaparelli was an Italian astronomer and science historian. He studied at the University of Turin and Berlin Observatory. In 1859-1860 he worked in Pulkovo Observatory and then worked for over forty years at Brera Observatory...

 (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
Adolphe Quetelet
Lambert Adolphe Jacques Quetelet was a Belgian astronomer, mathematician, statistician and sociologist. He founded and directed the Brussels Observatory and was influential in introducing statistical methods to the social sciences...

 (1853), Richard Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...

 (1860), Helmert (1872), Hermann Laurent (1873), Liagre, Didion, and Karl Pearson
Karl Pearson
Karl Pearson FRS was an influential English mathematician who has been credited for establishing the disciplineof mathematical statistics....

. Augustus De Morgan
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....

 and George Boole
George Boole
George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...

 improved the exposition of the theory.

Andrey Markov
Andrey Markov
Andrey Andreyevich Markov was a Russian mathematician. He is best known for his work on theory of stochastic processes...

 introduced the notion of Markov chains (1906), which played an important role in stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

es theory and its applications. The modern theory of probability based on the measure theory
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

 was developed by Andrey Kolmogorov
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...

 (1931).

On the geometric side (see integral geometry
Integral geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant transformations from the space of functions on one geometrical space to the...

) contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin
Artemas Martin
Artemas Martin was a self-educated American mathematician.-Biography:Martin was born on August 3, 1835 in Steuben County, New York and grew up in Venango County, Pennsylvania...

).


Theory



Like other theories
Theory
The English word theory was derived from a technical term in Ancient Greek philosophy. The word theoria, , meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action...

, the theory of probability
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 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 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
Richard Threlkeld Cox
Richard Threlkeld Cox was a professor of physics at Johns Hopkins University, known for Cox's theorem relating to the foundations of probability....

 formulation. In Kolmogorov's formulation (see probability space
Probability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...

), sets are interpreted as event
Event (probability theory)
In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...

s and probability itself as a measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

 on a class of sets. In Cox's theorem
Cox's theorem
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. This derivation justifies the so-called "logical" interpretation of probability. As the laws of probability derived by Cox's theorem are applicable to...

, 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
Probability axioms
In probability theory, the probability P of some event E, denoted P, is usually defined in such a way that P satisfies the Kolmogorov axioms, named after Andrey Kolmogorov, which are described below....

 are the same, except for technical details.

There are other methods for quantifying uncertainty,
such as the Dempster-Shafer theory
Dempster-Shafer theory
The Dempster–Shafer theory is a mathematical theory of evidence. It allows one to combine evidence from different sources and arrive at a degree of belief that takes into account all the available evidence. The theory was first developed by Arthur P...

 or possibility theory
Possibility theory
Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. D. Dubois and H. Prade further...

,
but those are essentially different and not compatible with the laws of probability as already understood.

Applications


Probability theory is applied in everyday life in risk
Risk
Risk is the potential that a chosen action or activity will lead to a loss . The notion implies that a choice having an influence on the outcome exists . Potential losses themselves may also be called "risks"...

 assessment and in trade on commodity markets
Commodity markets
Commodity markets are markets where raw or primary products are exchanged. These raw commodities are traded on regulated commodities exchanges, in which they are bought and sold in standardized contracts....

. Governments typically apply probabilistic methods in environmental regulation, where it is called pathway analysis.
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
Behavioral finance
Behavioral economics and its related area of study, behavioral finance, use social, cognitive and emotional factors in understanding the economic decisions of individuals and institutions performing economic functions, including consumers, borrowers and investors, and their effects on market...

 emerged to describe the effect of such groupthink
Groupthink
Groupthink is a psychological phenomenon that occurs within groups of people. It is the mode of thinking that happens when the desire for harmony in a decision-making group overrides a realistic appraisal of alternatives. Group members try to minimize conflict and reach a consensus decision without...

 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 profoundly affected 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
Democracy is generally defined as a form of government in which all adult citizens have an equal say in the decisions that affect their lives. Ideally, this includes equal participation in the proposal, development and passage of legislation into law...

.

Another significant application of probability theory in everyday life is reliability
Reliability theory of aging and longevity
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 a general theory of systems failure, known as reliability theory.-Overview:...

. Many consumer products, such as automobiles and consumer electronics, use reliability theory
Reliability theory
Reliability theory describes the probability of a system completing its expected function during an interval of time. It is the basis of reliability engineering, which is an area of study focused on optimizing the reliability, or probability of successful functioning, of systems, such as airplanes,...

 in product design to reduce the probability of failure. Failure probability may influence a manufacture's decisions on a product's warranty
Warranty
In business and legal transactions, a warranty is an assurance by one party to the other party that specific facts or conditions are true or will happen; the other party is permitted to rely on that assurance and seek some type of remedy if it is not true or followed.In real estate transactions, a...

.

The cache language model
Cache language model
A cache language model is a type of statistical language model. These occur in the natural language processing subfield of computer science and assign probabilities to given sequences of words by means of a probability distribution...

 and other statistical language models that are used in natural language processing
Natural language processing
Natural language processing is a field of computer science and linguistics concerned with the interactions between computers and human languages; it began as a branch of artificial intelligence....

 are also examples of applications of probability theory.

Mathematical treatment



Consider an experiment that can produce a number of results. The collection of all results is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling a die can produce six possible results. One collection of possible results give an odd number on the die. Thus, the subset {1,3,5} is an element of the power set of the sample space of die rolls. These collections are called "events." In this case, {1,3,5} is the event that the die falls on some odd number. If the results that actually occur fall in a given event, the event is said to have occurred.

A probability is a way of assigning
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 every event a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) is assigned a value of one. To qualify as a probability, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events (events with no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events.

The probability of an event
Event (probability theory)
In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...

 A is written as P(A), p(A) or Pr(A). This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using the concept of a measure.

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 . As an example, the chance of not rolling a six on a six-sided die is . See Complementary event
Complementary event
In probability theory, the complement of any event A is the event [not A], i.e. the event that A does not occur. The event A and its complement [not A] are mutually exclusive and exhaustive. Generally, there is only one event B such that A and B are both mutually exclusive and...

 for a more complete treatment.

If both events A and B occur on a single performance of an experiment, this is called the intersection or joint probability
Joint distribution
In the study of probability, given two random variables X and Y that are defined on the same probability space, the joint distribution for X and Y defines the probability of events defined in terms of both X and Y...

 of A and B, denoted as .

Independent Probability


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

Mutually Exclusive


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

Not Mutually Exclusive


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


Conditional probability
Conditional probability
In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...

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. Note that in this case A and B are independent.

Summary of Probabilities

Summary of probabilities
EventProbability
A
not A
A or B
A and B
A given B

Relation to randomness



In a deterministic
Determinism
Determinism is the general philosophical thesis that states that for everything that happens there are conditions such that, given them, nothing else could happen. There are many versions of this thesis. Each of them rests upon various alleged connections, and interdependencies of things and...

 universe, based on Newtonian concepts, there would be no probability if all conditions are known, (Laplace's demon
Laplace's demon
In the history of science, Laplace's demon was the first published articulation of causal or scientific determinism by Pierre-Simon Laplace in 1814...

). In the case of a roulette wheel, if the force of the hand and the period of that force are known, 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
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

 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.

Probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 is required to describe nature. A revolutionary discovery of early 20th century physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

 was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

. The objective wave function evolves deterministically but, according to the Copenhagen interpretation
Copenhagen interpretation
The Copenhagen interpretation is one of the earliest and most commonly taught interpretations of quantum mechanics. It holds that quantum mechanics does not yield a description of an objective reality but deals only with probabilities of observing, or measuring, various aspects of energy quanta,...

, randomness is explained by a wave function collapse when an observation is made. However, the loss of determinism
Determinism
Determinism is the general philosophical thesis that states that for everything that happens there are conditions such that, given them, nothing else could happen. There are many versions of this thesis. Each of them rests upon various alleged connections, and interdependencies of things and...

 for the sake of instrumentalism
Instrumentalism
In the philosophy of science, instrumentalism is the view that a scientific theory is a useful instrument in understanding the world. A concept or theory should be evaluated by how effectively it explains and predicts phenomena, as opposed to how accurately it describes objective...

 did not meet with universal approval. Albert Einstein
Albert Einstein
Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

 famously remarked in a letter to Max Born
Max Born
Max Born was a German-born physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a number of notable physicists in the 1920s and 30s...

: "I am convinced that God does not play dice". Like Einstein, Erwin Schrödinger
Erwin Schrödinger
Erwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

, who discovered the wave function, believed quantum mechanics is a statistical approximation of an underlying deterministic reality. In modern interpretations, quantum decoherence
Quantum decoherence
In quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior...

 accounts for subjectively probabilistic behavior.

Quotations

  • Damon Runyon
    Damon Runyon
    Alfred Damon Runyon was an American newspaperman and writer.He was best known for his short stories celebrating the world of Broadway in New York City that grew out of the Prohibition era. To New Yorkers of his generation, a "Damon Runyon character" evoked a distinctive social type from the...

    , "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
    Pierre-Simon Laplace
    Pierre-Simon, marquis de Laplace was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste...

     "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).

External links