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In common parlance, randomness is the apparent or actual lack of
pattern A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of pattern formed of geometric Geometry (from the grc, ...

pattern
or
predictability Predictability is the degree to which a correct prediction or forecasting, forecast of a system's Classical mechanics, state can be made, either qualitatively or quantitatively. Predictability and causality Causal determinism has a strong relati ...
in events. A random sequence of events,
symbol A symbol is a mark, sign, or word In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (linguistics), m ...

symbol
s or steps often has no
order Order, ORDER or Orders may refer to: * Orderliness Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the
probability distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
is known, the frequency of different outcomes over repeated events (or "trials") is predictable.Strictly speaking, the frequency of an outcome will converge
almost surely In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
to a predictable value as the number of trials becomes arbitrarily large. Non-convergence or convergence to a different value is possible, but has
probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

probability
zero.
For example, when throwing two
dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random numbers, commonly as part of tabletop game Tabletop games are game with separate sliding d ...

dice
, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance,
probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

probability
, and
information entropy In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent in the variable's possible outcomes. The concept of information entropy was introduced by Claude Shannon in hi ...
. The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a
random variable A random variable is a variable whose values depend on outcomes of a random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...
is an assignment of a numerical value to each possible outcome of an
event space In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in
random sequence The concept of a random sequence is essential in probability theory and statistics. The concept generally relies on the notion of a sequence of random variables and many statistical discussions begin with the words "let ''X''1,...,''Xn'' be independ ...
s. A
random process In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
is a sequence of random variables whose outcomes do not follow a
deterministic Determinism is the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental reality Reality is the ...
pattern, but follow an evolution described by
probability distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
s. These and other constructs are extremely useful in
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
and the various
applications of randomness Randomness In common parlance, randomness is the apparent or actual lack of pattern A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A ...
. Randomness is most often used in
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

statistics
to signify well-defined statistical properties.
Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
s, which rely on random input (such as from
random number generators Random number generation is a process which, often by means of a random number generator (RNG), generates a sequence of numbers or symbols that cannot be reasonably predicted better than by a random chance. Random number generators can be truly ra ...
or
pseudorandom number generator A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of proble ...
s), are important techniques in science, particularly in the field of
computational science Computational science, also known as scientific computing or scientific computation (SC), is a field that uses advanced computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes th ...
. By analogy,
quasi-Monte Carlo method In numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathemat ...
s use quasi-random number generators. Random selection, when narrowly associated with a
simple random sample In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more ...
, is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. Note that a random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, say research subjects, has the same probability of being chosen, then we can say the selection process is random. According to
Ramsey theory Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related s ...
, pure randomness is impossible, especially for large structures. Mathematician
Theodore Motzkin Theodore Samuel Motzkin (26 March 1908 – 15 December 1970) was an Israel Israel (; he, יִשְׂרָאֵל, translit=Yīsrāʾēl; ar, إِسْرَائِيل, translit=ʾIsrāʾīl), officially the State of Israel ( he, מְדִינ ...
suggested that "while disorder is more probable in general, complete disorder is impossible". Misunderstanding this can lead to numerous
conspiracy theories A conspiracy theory is an explanation for an event or situation that invokes a conspiracy A conspiracy, also known as a plot, is a secret plan or agreement between persons (called conspirers or conspirators) for an unlawful or harmful purp ...
. Cristian S. Calude stated that "given the impossibility of true randomness, the effort is directed towards studying degrees of randomness". Cristian S. Calude, (2017)
"Quantum Randomness: From Practice to Theory and Back"
in "The Incomputable Journeys Beyond the Turing Barrier" Editors:
S. Barry Cooper S. Barry Cooper (9 October 1943 – 26 October 2015) was an English mathematician and computability theorist. He was a professor of Pure Mathematics Pure mathematics is the study of mathematical concepts independently of any application outside ...
, Mariya I. Soskova, 169–181, doi:10.1007/978-3-319-43669-2_11.
It can be proven that there is infinite hierarchy (in terms of quality or strength) of forms of randomness.


History

In ancient history, the concepts of chance and randomness were intertwined with that of fate. Many ancient peoples threw
dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random numbers, commonly as part of tabletop game Tabletop games are game with separate sliding d ...

dice
to determine fate, and this later evolved into games of chance. Most ancient cultures used various methods of
divination Divination (from Latin ''divinare'', 'to foresee, to foretell, to predict, to prophesy') is the attempt to gain insight into a question or situation by way of an occult The occult, in the broadest sense, is a category of supernatural ...

divination
to attempt to circumvent randomness and fate. The Chinese of 3000 years ago were perhaps the earliest people to formalize odds and chance. The Greek philosophers discussed randomness at length, but only in non-quantitative forms. It was only in the 16th century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

calculus
had a positive impact on the formal study of randomness. In the 1888 edition of his book ''The Logic of Chance'',
John Venn John Venn, FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family Resources ...

John Venn
wrote a chapter on ''The conception of randomness'' that included his view of the randomness of the digits of , by using them to construct a
random walk In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
in two dimensions. The early part of the 20th century saw a rapid growth in the formal analysis of randomness, as various approaches to the mathematical foundations of probability were introduced. In the mid-to-late-20th century, ideas of
algorithmic information theory Algorithmic information theory (AIT) is a branch of theoretical computer science Theoretical computer science (TCS) is a subset of general computer science Computer science deals with the theoretical foundations of information, algorith ...
introduced new dimensions to the field via the concept of
algorithmic randomness Intuitively, an algorithmically random sequence (or random sequenceThe concept of a random sequence is essential in probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several differe ...
. Although randomness had often been viewed as an obstacle and a nuisance for many centuries, in the 20th century computer scientists began to realize that the ''deliberate'' introduction of randomness into computations can be an effective tool for designing better algorithms. In some cases, such
randomized algorithms A randomized algorithm is an algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are used as specif ...
even outperform the best deterministic methods.


In science

Many scientific fields are concerned with randomness: *
Algorithmic probability In algorithmic information theory Algorithmic information theory (AIT) is a branch of theoretical computer science An artistic representation of a Turing machine. Turing machines are used to model general computing devices. Theoretical computer ...
*
Chaos theory Chaos theory is an interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from several other fields ...
*
Cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia ''-logy'' is a suffix in the English language, used with words originally adapted from Ancient Greek ending in (''- ...

Cryptography
*
Game theory Game theory is the study of mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. ...
*
Information theory Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of Digital data, digital information. The field was fundamentally established by the ...
*
Pattern recognition Pattern recognition is the automated recognition of pattern A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric pattern is a kind of ...
*
Probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
*
Quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
*
Statistical mechanics In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
*
Statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

Statistics


In the physical sciences

In the 19th century, scientists used the idea of random motions of molecules in the development of
statistical mechanics In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
to explain phenomena in
thermodynamics Thermodynamics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ot ...

thermodynamics
and the properties of gases. According to several standard interpretations of
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
, microscopic phenomena are objectively random. That is, in an experiment that controls all causally relevant parameters, some aspects of the outcome still vary randomly. For example, if a single unstable
atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of ato ...

atom
is placed in a controlled environment, it cannot be predicted how long it will take for the atom to decay—only the probability of decay in a given time. Thus, quantum mechanics does not specify the outcome of individual experiments, but only the probabilities.
Hidden variable theories In physics, hidden-variable theories are proposals to provide explanations of quantum mechanical phenomena through the introduction of unobservable hypothetical entities. The existence of fundamental indeterminacy for some measurements is assu ...
reject the view that nature contains irreducible randomness: such theories posit that in the processes that appear random, properties with a certain statistical distribution are at work behind the scenes, determining the outcome in each case.


In biology

The modern evolutionary synthesis ascribes the observed diversity of life to random genetic
mutation In biology Biology is the natural science that studies life and living organisms, including their anatomy, physical structure, Biochemistry, chemical processes, Molecular biology, molecular interactions, Physiology, physiological mechan ...
s followed by
natural selection Natural selection is the differential survival and reproduction of individuals due to differences in phenotype right , Here the relation between genotype and phenotype is illustrated, using a Punnett square, for the character of peta ...
. The latter retains some random mutations in the
gene pool The gene pool is the set of all genes, or genetic information, in any population, usually of a particular species. Description A large gene pool indicates extensive genetic diversity, which is associated with robust populations that can surviv ...
due to the systematically improved chance for survival and reproduction that those mutated genes confer on individuals who possess them. Several authors also claim that evolution (and sometimes development) requires a specific form of randomness, namely the introduction of qualitatively new behaviors. Instead of the choice of one possibility among several pre-given ones, this randomness corresponds to the formation of new possibilities. The characteristics of an organism arise to some extent deterministically (e.g., under the influence of genes and the environment), and to some extent randomly. For example, the ''density'' of
freckles Freckles are clusters of concentrated melaninized cells which are most easily visible on people with a fair complexion. Freckles do not have an increased number of the melanin-producing cells, or melanocytes, but instead have melanocytes th ...

freckles
that appear on a person's skin is controlled by genes and exposure to light; whereas the exact location of ''individual'' freckles seems random. As far as behavior is concerned, randomness is important if an animal is to behave in a way that is unpredictable to others. For instance, insects in flight tend to move about with random changes in direction, making it difficult for pursuing predators to predict their trajectories.


In mathematics

The mathematical theory of
probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...

probability
arose from attempts to formulate mathematical descriptions of chance events, originally in the context of
gambling Gambling (also known as betting) is the wagering something of Value (economics), value ("the stakes") on an Event (probability theory), event with an uncertain outcome with the intent of winning something else of value. Gambling thus requires ...
, but later in connection with physics.
Statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

Statistics
is used to infer the underlying
probability distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
of a collection of empirical observations. For the purposes of
simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of models; the model represents the key characteristics or behaviors of the selected system or process, whereas the simulat ...

simulation
, it is necessary to have a large supply of random numbers—or means to generate them on demand.
Algorithmic information theory Algorithmic information theory (AIT) is a branch of theoretical computer science Theoretical computer science (TCS) is a subset of general computer science Computer science deals with the theoretical foundations of information, algorith ...
studies, among other topics, what constitutes a
random sequence The concept of a random sequence is essential in probability theory and statistics. The concept generally relies on the notion of a sequence of random variables and many statistical discussions begin with the words "let ''X''1,...,''Xn'' be independ ...
. The central idea is that a string of
bit The bit is a basic unit of information in computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algo ...
s is random if and only if it is shorter than any computer program that can produce that string ( Kolmogorov randomness), which means that random strings are those that cannot be
compressed Compression may refer to: Physical science *Compression (physics), size reduction due to forces *Compression member, a structural element such as a column *Compressibility, susceptibility to compression *Gas compression *Compression ratio, of a co ...
. Pioneers of this field include
Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovie ...
and his student
Per Martin-Löf Per Erik Rutger Martin-Löf (; ; born 8 May 1942) is a Swedish Swedish or ' may refer to: * Anything from or related to Sweden, a country in Northern Europe * Swedish language, a North Germanic language spoken primarily in Sweden and Finland * S ...
,
Ray Solomonoff Ray Solomonoff (July 25, 1926 – December 7, 2009) was the inventor of algorithmic probability, his General Theory of Inductive Inference (also known as Universal Inductive Inference),Samuel Rathmanner and Marcus Hutter. A philosophical treatise of ...
, and
Gregory Chaitin Gregory John Chaitin ( ; born 25 June 1947) is an Argentine Argentines (also known as Argentinians or Argentineans; es, masculine argentinos; feminine Femininity (also called womanliness or girlishness) is a set of attributes, beha ...
. For the notion of infinite sequence, mathematicians generally accept
Per Martin-Löf Per Erik Rutger Martin-Löf (; ; born 8 May 1942) is a Swedish Swedish or ' may refer to: * Anything from or related to Sweden, a country in Northern Europe * Swedish language, a North Germanic language spoken primarily in Sweden and Finland * S ...
's semi-eponymous definition: An infinite sequence is random if and only if it withstands all recursively enumerable null sets. The other notions of random sequences include, among others, recursive randomness and Schnorr randomness, which are based on recursively computable martingales. It was shown by
Yongge Wang Yongge Wang (born 1967) is a computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer scie ...
that these randomness notions are generally different. Randomness occurs in numbers such as log(2) and . The decimal digits of pi constitute an infinite sequence and "never repeat in a cyclical fashion." Numbers like pi are also considered likely to be normal:


In statistics

In statistics, randomness is commonly used to create simple random samples. This allows surveys of completely random groups of people to provide realistic data that is reflective of the population. Common methods of doing this include drawing names out of a hat or using a random digit chart (a large table of random digits).


In information science

In information science, irrelevant or meaningless data is considered noise. Noise consists of numerous transient disturbances, with a statistically randomized time distribution. In
communication theory A communication theory is a proposed description of communication phenomena, the relationships among them, a storyline describing these relationships, and an argument for these three elements. Communication theory provides a way of talking about ...

communication theory
, randomness in a signal is called "noise", and is opposed to that component of its variation that is causally attributable to the source, the signal. In terms of the development of random networks, for communication randomness rests on the two simple assumptions of
Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a renowned Hungarian mathematician In this page we keep the names in Hungarian order (family name first). {{compact ToC , short1, side=yes A * Alexits György (1899–1 ...

Paul Erdős
and Alfréd Rényi, who said that there were a fixed number of nodes and this number remained fixed for the life of the network, and that all nodes were equal and linked randomly to each other.


In finance

The
random walk hypothesisThe random walk hypothesis is a financial theory stating that stock market A stock market, equity market, or share market is the aggregation of buyers and sellers of stocks (also called shares), which represent ownership claims on busine ...
considers that asset prices in an organized
market Market may refer to: *Market (economics) *Market economy *Marketplace, a physical marketplace or public market Geography *Märket, an island shared by Finland and Sweden Art, entertainment, and media Films *Market (1965 film), ''Market'' (1965 ...
evolve at random, in the sense that the expected value of their change is zero but the actual value may turn out to be positive or negative. More generally, asset prices are influenced by a variety of unpredictable events in the general economic environment.


In politics

Random selection can be an official method to resolve tied elections in some jurisdictions. Its use in politics originates long ago. Many offices in
Ancient Athens Athens , image_skyline = File:Athens Montage L.png, center, 275px, alt=Athens montage. Clicking on an image in the picture causes the browser to load the appropriate article. rect 15 15 985 460 Acropolis of Athens rec ...
were chosen by lot instead of modern voting.


Randomness and religion

Randomness can be seen as conflicting with the
deterministic Determinism is the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental reality Reality is the ...

deterministic
ideas of some religions, such as those where the universe is created by an omniscient deity who is aware of all past and future events. If the universe is regarded to have a purpose, then randomness can be seen as impossible. This is one of the rationales for religious opposition to
evolution Evolution is change in the heritable Heredity, also called inheritance or biological inheritance, is the passing on of Phenotypic trait, traits from parents to their offspring; either through asexual reproduction or sexual reproduction, ...

evolution
, which states that
non-random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. ...
selection is applied to the results of random genetic variation.
Hindu Hindus (; ) are persons who regard themselves as culturally, ethnically, or religiously adhering to aspects of Hinduism Hinduism () is an Indian religion Indian religions, sometimes also termed Dharmic religions or Indic re ...

Hindu
and
Buddhist Buddhism (, ) is the world's fourth-largest religion Religion is a social Social organisms, including humans, live collectively in interacting populations. This interaction is considered social whether they are aware of it or not, an ...

Buddhist
philosophies state that any event is the result of previous events, as is reflected in the concept of
karma Karma (; sa, कर्म}, ; pi, kamma, italic=yes) means action, work, or deed. For the believers in spirituality the term also refers to the Spirituality, spiritual principle of cause and effect, often descriptively called the principl ...

karma
. As such, this conception is at odd with the idea of randomness, and any reconciliation between both of them would require an explanation. In some religious contexts, procedures that are commonly perceived as randomizers are used for divination.
Cleromancy Cleromancy is a form of sortition, casting of wikt:lot#Noun, lots, in which an outcome is determined by means that normally would be considered random, such as the rolling of dice, but that are sometimes believed to reveal the will of God, or othe ...
uses the casting of bones or dice to reveal what is seen as the will of the gods.


Applications

In most of its mathematical, political, social and religious uses, randomness is used for its innate "fairness" and lack of bias. Politics:
Athenian democracy Athenian democracy developed around the 6th century BC in the Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located ...
was based on the concept of
isonomia ''Isonomia'' (ἰσονομία "equality of political rights Civil and political rights are a class of rights that protect individuals' political freedom, freedom from infringement by governments, social organizations, and private individuals. ...
(equality of political rights), and used complex allotment machines to ensure that the positions on the ruling committees that ran Athens were fairly allocated. Allotment is now restricted to selecting jurors in Anglo-Saxon legal systems, and in situations where "fairness" is approximated by
randomizationRandomization is the process of making something random; in various contexts this involves, for example: * generating a random permutation of a sequence (such as when shuffle, shuffling cards); * selecting a random sample of a population (important i ...
, such as selecting
juror A jury is a sworn body of people (the jurors) convened to render an impartiality, impartial verdict (a Question of fact, finding of fact on a question) officially submitted to them by a court, or to set a sentence (law), penalty or Judgment (la ...
s and military
draft Draft, The Draft, or Draught may refer to: Watercraft dimensions * Draft (hull), the distance from waterline to keel of a vessel * Draft (sail), degree of curvature in a sail * Air draft, distance from waterline to the highest point on a vessel ...

draft
lotteries. Games: Random numbers were first investigated in the context of
gambling Gambling (also known as betting) is the wagering something of Value (economics), value ("the stakes") on an Event (probability theory), event with an uncertain outcome with the intent of winning something else of value. Gambling thus requires ...
, and many randomizing devices, such as
dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random numbers, commonly as part of tabletop game Tabletop games are game with separate sliding d ...

dice
, shuffling playing cards, and
roulette Roulette is a casino A casino is a facility for certain types of gambling. Casinos are often built near or combined with hotels, resorts, restaurants, retail shopping, cruise ships, and other tourist attractions. Some casinos are also known ...

roulette
wheels, were first developed for use in gambling. The ability to produce random numbers fairly is vital to electronic gambling, and, as such, the methods used to create them are usually regulated by government Gaming Control Boards. Random drawings are also used to determine
lottery A lottery is a form of gambling Gambling (also known as betting) is the wagering something of Value (economics), value ("the stakes") on an Event (probability theory), event with an uncertain outcome with the intent of winning something e ...
winners. In fact, randomness has been used for games of chance throughout history, and to select out individuals for an unwanted task in a fair way (see
drawing strawsDrawing straws is a selection method, or a form of sortition In governance Governance comprises all of the processes of governing – whether undertaken by the government of a state (polity), state, by a market (economics), market, or by a so ...
). Sports: Some sports, including
American football American football, referred to simply as football in the United States and Canada and also known as gridiron, is a team sport played by two teams of eleven players on a rectangular American football field, field with goalposts at each end. ...

American football
, use
coin toss A coin is a small, flat, (usually, depending on the country or value) round piece of metal A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fr ...
es to randomly select starting conditions for games or
seed A seed is an embryonic ''Embryonic'' is the twelfth studio album by experimental rock band the Flaming Lips released on October 13, 2009, on Warner Bros. Records, Warner Bros. The band's first double album, it was released to generally positi ...
tied teams for postseason play. The
National Basketball Association The National Basketball Association (NBA) is a professional basketball sports league, league in North America. The league is composed of 30 teams (29 in the United States and 1 in Canada) and is one of the four major professional sports leagu ...
uses a weighted
lottery A lottery is a form of gambling Gambling (also known as betting) is the wagering something of Value (economics), value ("the stakes") on an Event (probability theory), event with an uncertain outcome with the intent of winning something e ...
to order teams in its draft. Mathematics: Random numbers are also employed where their use is mathematically important, such as sampling for
opinion poll An opinion poll, often simply referred to as a poll or a survey, is a survey (human research), human research survey of public opinion from a particular sampling (statistics), sample. Opinion polls are usually designed to represent the opinions o ...
s and for statistical sampling in
quality control Quality control (QC) is a process by which entities review the quality of all factors involved in production. ISO 9000 The ISO 9000 family of quality management systems A quality management system (QMS) is a collection of business process ...

quality control
systems. Computational solutions for some types of problems use random numbers extensively, such as in the
Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determini ...
and in
genetic algorithm spacecraft antenna. This complicated shape was found by an evolutionary computer design program to create the best radiation pattern. It is known as an evolved antenna. In computer science Computer science deals with the theoretical found ...

genetic algorithm
s. Medicine: Random allocation of a clinical intervention is used to reduce bias in controlled trials (e.g.,
randomized controlled trials A randomized controlled trial (or randomized control trial; RCT) is a type of '' scientific experiment'' (e.g. a ''clinical trial Clinical trials are experiments or observations done in clinical research. Such prospective biomedical or behavio ...
). Religion: Although not intended to be random, various forms of
divination Divination (from Latin ''divinare'', 'to foresee, to foretell, to predict, to prophesy') is the attempt to gain insight into a question or situation by way of an occult The occult, in the broadest sense, is a category of supernatural ...

divination
such as
cleromancy Cleromancy is a form of sortition, casting of wikt:lot#Noun, lots, in which an outcome is determined by means that normally would be considered random, such as the rolling of dice, but that are sometimes believed to reveal the will of God, or othe ...
see what appears to be a random event as a means for a divine being to communicate their will (see also
Free will Free will is the capacity of agents to choose between different possible courses of action ACTION is a bus operator in , Australia owned by the . History On 19 July 1926, the commenced operating public bus services between Eastlake ( ...

Free will
and
Determinism Determinism is the philosophical Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical or mental reality Reality is the ...
for more).


Generation

It is generally accepted that there exist three mechanisms responsible for (apparently) random behavior in systems: # ''Randomness'' coming from the environment (for example,
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physica ...

Brownian motion
, but also
hardware random number generator In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithm of an algorithm (Euclid's algorithm) for calculating the greatest comm ...
s). # ''Randomness'' coming from the initial conditions. This aspect is studied by
chaos theory Chaos theory is an interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from several other fields ...
, and is observed in systems whose behavior is very sensitive to small variations in initial conditions (such as
pachinko is a type of mechanical game originating in Japan Japan ( ja, 日本, or , and formally ) is an in . It is situated in the northwest , and is bordered on the west by the , while extending from the in the north toward the and in ...

pachinko
machines and
dice Dice (singular die or dice) are small, throwable objects with marked sides that can rest in multiple positions. They are used for generating random numbers, commonly as part of tabletop game Tabletop games are game with separate sliding d ...

dice
). # ''Randomness'' intrinsically generated by the system. This is also called
pseudorandomness Pseudorandomness measures the extent to which a sequence of numbers, though produced by a completely deterministic and repeatable process, appear to be patternless. The pattern's seeming randomness is the crux of much online and other security ...
, and is the kind used in
pseudo-random number generator Pseudorandomness measures the extent to which a sequence of numbers, though produced by a completely deterministic Determinism is the philosophical view that all events are determined completely by previously existing causes. Deterministic ...
s. There are many algorithms (based on
arithmetics Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cra ...
or
cellular automaton A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation A model is an informative representation of an object, person or system. The term originally denoted the plan A plan is typically any diagram or list ...

cellular automaton
) for generating pseudorandom numbers. The behavior of the system can be determined by knowing the seed state and the algorithm used. These methods are often quicker than getting "true" randomness from the environment. The many
applications of randomness Randomness In common parlance, randomness is the apparent or actual lack of pattern A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A ...
have led to many different methods for generating random data. These methods may vary as to how unpredictable or
statistically randomA numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of an ideal dice, dice roll or the digits of pi, π exhibit statistical randomness. Statistical randomness ...
they are, and how quickly they can generate random numbers. Before the advent of computational
random number generator Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual languag ...
s, generating large amounts of sufficiently random numbers (which is important in statistics) required a lot of work. Results would sometimes be collected and distributed as
random number table Random number tables have been used in statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social probl ...
s.


Measures and tests

There are many practical measures of randomness for a binary sequence. These include measures based on frequency,
discrete transform In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image processing, images, and scientific measurements. Signal pr ...
s,
complexity Complexity characterises the behaviour of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environme ...
, or a mixture of these, such as the tests by Kak, Phillips, Yuen, Hopkins, Beth and Dai, Mund, and Marsaglia and Zaman.
Quantum nonlocality In theoretical physics, quantum nonlocality refers to the phenomenon by which the Measurement in quantum mechanics, measurement statistics of a multipartite quantum system do not admit an interpretation in terms of a Local realism, local realistic ...
has been used to certify the presence of genuine or strong form of randomness in a given string of numbers.


Misconceptions and logical fallacies

Popular perceptions of randomness are frequently mistaken, and are often based on fallacious reasoning or intuitions.


Fallacy: a number is "due"

This argument is, "In a random selection of numbers, since all numbers eventually appear, those that have not come up yet are 'due', and thus more likely to come up soon." This logic is only correct if applied to a system where numbers that come up are removed from the system, such as when
playing card A playing card is a piece of specially prepared card stock, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a Pap ...
s are drawn and not returned to the deck. In this case, once a jack is removed from the deck, the next draw is less likely to be a jack and more likely to be some other card. However, if the jack is returned to the deck, and the deck is thoroughly reshuffled, a jack is as likely to be drawn as any other card. The same applies in any other process where objects are selected independently, and none are removed after each event, such as the roll of a die, a coin toss, or most
lottery A lottery is a form of gambling Gambling (also known as betting) is the wagering something of Value (economics), value ("the stakes") on an Event (probability theory), event with an uncertain outcome with the intent of winning something e ...
number selection schemes. Truly random processes such as these do not have memory, which makes it impossible for past outcomes to affect future outcomes. In fact, there is no finite number of trials that can guarantee a success.


Fallacy: a number is "cursed" or "blessed"

In a random sequence of numbers, a number may be said to be cursed because it has come up less often in the past, and so it is thought that it will occur less often in the future. A number may be assumed to be blessed because it has occurred more often than others in the past, and so it is thought likely to come up more often in the future. This logic is valid only if the randomisation might be biased, for example if a die is suspected to be loaded then its failure to roll enough sixes would be evidence of that loading. If the die is known to be fair, then previous rolls can give no indication of future events. In nature, events rarely occur with a frequency that is known ''
a priori ''A priori'' and ''a posteriori'' ('from the earlier' and 'from the later', respectively) are Latin phrases used in philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaph ...
'', so observing outcomes to determine which events are more probable makes sense. However, it is fallacious to apply this logic to systems designed and known to make all outcomes equally likely, such as shuffled cards, dice, and roulette wheels.


Fallacy: odds are never dynamic

In the beginning of a scenario, one might calculate the probability of a certain event. However, as soon as one gains more information about the scenario, one may need to re-calculate the probability accordingly. For example, when being told that a woman has two children, one might be interested in knowing if either of them is a girl, and if yes, what is probability that the other child is also a girl. Considering the two events independently, one might expect that the probability that the other child is female is ½ (50%), but by building a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a space (mathematics), mathematical construct that provides a formal model of a randomness, random process or "experiment". For example, one can define a ...
illustrating all possible outcomes, one would notice that the probability is actually only ⅓ (33%). To be sure, the probability space does illustrate four ways of having these two children: boy-boy, girl-boy, boy-girl, and girl-girl. But once it is known that at least one of the children is female, this rules out the boy-boy scenario, leaving only three ways of having the two children: boy-girl, girl-boy, girl-girl. From this, it can be seen only ⅓ of these scenarios would have the other child also be a girl (see Boy or girl paradox for more). In general, by using a probability space, one is less likely to miss out on possible scenarios, or to neglect the importance of new information. This technique can be used to provide insights in other situations such as the
Monty Hall problem The Monty Hall problem is a brain teaser, in the form of a probability puzzle, loosely based on the American television game show ''Let's Make a Deal'' and named after its original host, Monty Hall. The problem was originally posed (and solved) in ...
, a game show scenario in which a car is hidden behind one of three doors, and two goats are hidden as
booby prize {{short description, Joke prize given in recognition of a terrible performance or last-place finish A booby prize is a joke prize A prize is an award An award, sometimes called a distinction, is something given to a recipient as a token of ...
s behind the others. Once the contestant has chosen a door, the host opens one of the remaining doors to reveal a goat, eliminating that door as an option. With only two doors left (one with the car, the other with another goat), the player must decide to either keep their decision, or to switch and select the other door. Intuitively, one might think the player is choosing between two doors with equal probability, and that the opportunity to choose another door makes no difference. However, an analysis of the probability spaces would reveal that the contestant has received new information, and that changing to the other door would increase their chances of winning.


See also

*
Aleatory Aleatoricism ( ''ey-lee-uh-TAWR-uh-siz-uhm, -TOR-, al-ee'') or aleatorism, the noun associated with the adjectival aleatory and aleatoric is a term popularised by the musical composer Pierre Boulez Pierre Louis Joseph Boulez CBE (; 26 March ...
*
Chaitin's constant In the computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of A ...
* Chance (disambiguation) * Frequency probability * Indeterminism * Nonlinear system * Probability interpretations *
Probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
* Pseudorandomness * Random.org—generates random numbers using atmospheric noise * Sortition


Notes


References


Further reading

* ''Randomness'' by Deborah J. Bennett. Harvard University Press, 1998. . * ''Random Measures, 4th ed.'' by Olav Kallenberg. Academic Press, New York, London; Akademie-Verlag, Berlin, 1986. . * ''The Art of Computer Programming. Vol. 2: Seminumerical Algorithms, 3rd ed.'' by Donald Knuth, Donald E. Knuth. Reading, MA: Addison-Wesley, 1997. . * ''Fooled by Randomness, 2nd ed.'' by Nassim Nicholas Taleb. Thomson Texere, 2004. . * ''Exploring Randomness'' by
Gregory Chaitin Gregory John Chaitin ( ; born 25 June 1947) is an Argentine Argentines (also known as Argentinians or Argentineans; es, masculine argentinos; feminine Femininity (also called womanliness or girlishness) is a set of attributes, beha ...
. Springer-Verlag London, 2001. . * ''Random'' by Kenneth Chan includes a "Random Scale" for grading the level of randomness. * ''The Drunkard’s Walk: How Randomness Rules our Lives'' by Leonard Mlodinow. Pantheon Books, New York, 2008. .


External links


QuantumLab
Quantum random number generator with single photons as interactive experiment.
HotBits
generates random numbers from radioactive decay.
QRBG
Quantum Random Bit Generator
QRNG
Fast Quantum Random Bit Generator
A Pseudorandom Number Sequence Test Program (Public Domain)''Dictionary of the History of Ideas'':
Chance
Computing a Glimpse of RandomnessChance versus Randomness
from the Stanford Encyclopedia of Philosophy {{Authority control Randomness, Cryptography Statistical randomness