![RandomBitmap](https://upload.wikimedia.org/wikipedia/commons/e/ec/RandomBitmap.png)
In common usage, 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 shapes and typically repeated l ...
or
predictability
Predictability is the degree to which a correct prediction or forecast of a system's state can be made, either qualitatively or quantitatively.
Predictability and causality
Causal determinism has a strong relationship with predictability. Perf ...
in events.
A random sequence of events,
symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
s or steps often has no
order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if the
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
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, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...
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 concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
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 values, commonly as part of tabletop games, including dice games, board games, role-playing g ...
, 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 concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
, and
information entropy
In information theory, the entropy of a random variable is the average level of "information", "surprise", or "uncertainty" inherent to the variable's possible outcomes. Given a discrete random variable X, which takes values in the alphabet \ ...
.
The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
is an assignment of a numerical value to each possible outcome of an
event space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
. 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 and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appe ...
is a sequence of random variables whose outcomes do not follow a
deterministic
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
pattern, but follow an evolution described by
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s. These and other constructs are extremely useful 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 expressing it through a set o ...
and the various
applications of randomness
Randomness has many uses in science, art, statistics, cryptography, gaming, gambling, and other fields. For example, random assignment in randomized controlled trials helps scientists to test hypotheses, and random numbers or pseudorandom number ...
.
Randomness is most often used in
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
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 determi ...
s, which rely on random input (such as from
random number generators
Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols that cannot be reasonably predicted better than by random chance is generated. This means that the particular ou ...
or
pseudorandom number generator
A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generate ...
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 in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disc ...
. By analogy,
quasi-Monte Carlo method
In numerical analysis, the quasi-Monte Carlo method is a method for numerical integration and solving some other problems using low-discrepancy sequences (also called quasi-random sequences or sub-random sequences). This is in contrast to the regu ...
s use
quasi-random number generators.
Random selection, when narrowly associated with a
simple random sample
In statistics, a simple random sample (or SRS) is a subset of individuals (a sample) chosen from a larger set (a population) in which a subset of individuals are chosen randomly, all with the same probability. It is a process of selecting a sample ...
, 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 that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a ...
, pure randomness is impossible, especially for large structures. Mathematician
Theodore Motzkin
Theodore Samuel Motzkin (26 March 1908 – 15 December 1970) was an Israeli-American mathematician.
Biography
Motzkin's father Leo Motzkin, a Ukrainian Jew, went to Berlin at the age of thirteen to study mathematics. He pursued university studi ...
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 by sinister and powerful groups, often political in motivation, when other explanations are more probable.Additional sources:
*
*
*
* The term has a nega ...
.
Cristian S. Calude
Cristian Sorin Calude (born 21 April 1952) is a Romanian-New Zealander
mathematician and computer scientist.
Biography
After graduating from the Vasile Alecsandri National College in Galați, he studied at the University of Bucharest, where he ...
stated that "given the impossibility of true randomness, the effort is directed towards studying degrees of randomness".
Cristian S. Calude
Cristian Sorin Calude (born 21 April 1952) is a Romanian-New Zealander
mathematician and computer scientist.
Biography
After graduating from the Vasile Alecsandri National College in Galați, he studied at the University of Bucharest, where he ...
, (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 theory, computability theorist. He was a professor of Pure Mathematics at the University of Leeds.
Early life and education
Cooper grew up in ...
, Mariya I. Soskova
Mariya is a variation of the feminine given name Maria.
People
* Mariya Abakumova (born 1986), Russian Olympic javelin thrower
* Mariya Agapova (born 1997), Kazakhstani mixed martial arts fighter
* Mariya Alyokhina (born 1988), Russian politica ...
, 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
![Pompeii - Osteria della Via di Mercurio - Dice Players](https://upload.wikimedia.org/wikipedia/commons/1/16/Pompeii_-_Osteria_della_Via_di_Mercurio_-_Dice_Players.jpg)
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 values, commonly as part of tabletop games, including dice games, board games, role-playing g ...
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 occultic, standardized process or ritual. Used in various forms throughout histor ...
to attempt to circumvent randomness and fate. Beyond
religion
Religion is usually defined as a social- cultural system of designated behaviors and practices, morals, beliefs, worldviews, texts, sanctified places, prophecies, ethics, or organizations, that generally relates humanity to supernatural, ...
and
games of chance
A game of chance is in contrast with a game of skill. It is a game whose outcome is strongly influenced by some randomizing device. Common devices used include dice, spinning tops, playing cards, roulette wheels, or numbered balls drawn from ...
, randomness has been attested for
sortition
In governance, sortition (also known as selection by lottery, selection by lot, allotment, demarchy, stochocracy, aleatoric democracy, democratic lottery, and lottocracy) is the selection of political officials as a random sample from a larger ...
since at least ancient
Athenian democracy
Athenian democracy developed around the 6th century BC in the Greek city-state (known as a polis) of Athens, comprising the city of Athens and the surrounding territory of Attica. Although Athens is the most famous ancient Greek democratic city- ...
in the form of a
kleroterion
A kleroterion ( grc, κληρωτήριον) was a randomization device used by the Athenian polis during the period of democracy to select citizens to the boule, to most state offices, to the nomothetai, and to court juries.
The kleroterion w ...
.
The formalization of odds and chance was perhaps earliest done by the Chinese of 3,000 years ago. 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 mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
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, Fellow of the Royal Society, FRS, Fellow of the Society of Antiquaries of London, FSA (4 August 1834 – 4 April 1923) was an English mathematician, logician and philosopher noted for introducing Venn diagrams, which are used in l ...
wrote a chapter on ''The conception of randomness'' that included his view of the randomness of the digits of
pi, by using them to construct a
random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
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 that concerns itself with the relationship between computation and information of computably generated objects (as opposed to stochastically generated), such as st ...
introduced new dimensions to the field via the concept of
algorithmic randomness
Intuitively, an algorithmically random sequence (or random sequence) is a sequence of binary digits that appears random to any algorithm running on a (prefix-free or not) universal Turing machine. The notion can be applied analogously to sequenc ...
.
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 that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performan ...
even outperform the best deterministic methods.
In science
Many scientific fields are concerned with randomness:
*
Algorithmic probability In algorithmic information theory, algorithmic probability, also known as Solomonoff probability, is a mathematical method of assigning a prior probability to a given observation. It was invented by Ray Solomonoff in the 1960s.
It is used in induct ...
*
Chaos theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
*
Cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
*
Game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
*
Information theory
Information theory is the scientific study of the quantification (science), quantification, computer data storage, storage, and telecommunication, communication of information. The field was originally established by the works of Harry Nyquist a ...
*
Pattern recognition
Pattern recognition is the automated recognition of patterns and regularities in data. It has applications in statistical data analysis, signal processing, image analysis, information retrieval, bioinformatics, data compression, computer graphi ...
*
Percolation theory
In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected ...
*
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 expressing it through a set o ...
*
Quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
*
Random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
*
Statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
*
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
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, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
to explain phenomena in
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
and
the properties of gases.
According to several standard interpretations of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, 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
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
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 (possibly unobservable) hypothetical entities. The existence of fundamental indeterminacy for some measurem ...
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
Modern synthesis or modern evolutionary synthesis refers to several perspectives on evolutionary biology, namely:
* Modern synthesis (20th century), the term coined by Julian Huxley in 1942 to denote the synthesis between Mendelian genetics and s ...
ascribes the observed diversity of life to random genetic
mutation
In biology, a mutation is an alteration in the nucleic acid sequence of the genome of an organism, virus, or extrachromosomal DNA. Viral genomes contain either DNA or RNA. Mutations result from errors during DNA or viral replication, mi ...
s followed by
natural selection
Natural selection is the differential survival and reproduction of individuals due to differences in phenotype. It is a key mechanism of evolution, the change in the heritable traits characteristic of a population over generations. Charle ...
. 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 surv ...
due to the systematically improved chance for survival and reproduction that those mutated genes confer on individuals who possess them. The location of the mutation is not entirely random however as e.g. biologically important regions may be more protected from mutations.
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 that ...
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 concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
arose from attempts to formulate mathematical descriptions of chance events, originally in the context of
gambling
Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three el ...
, but later in connection with physics.
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
is used to infer the underlying
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
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 Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...
, 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 that concerns itself with the relationship between computation and information of computably generated objects (as opposed to stochastically generated), such as st ...
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 the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s is random if and only if it is shorter than any computer program that can produce that string (
Kolmogorov randomness
In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program (in a predetermined programming language) that produ ...
), which means that random strings are those that cannot be
compressed. 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 Sovi ...
and his student
Per Martin-Löf
Per Erik Rutger Martin-Löf (; ; born 8 May 1942) is a Swedish logician, philosopher, and mathematical statistician. He is internationally renowned for his work on the foundations of probability, statistics, mathematical logic, and computer scie ...
,
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 o ...
, and
Gregory Chaitin
Gregory John Chaitin ( ; born 25 June 1947) is an Argentine-American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer-t ...
. For the notion of infinite sequence, mathematicians generally accept
Per Martin-Löf
Per Erik Rutger Martin-Löf (; ; born 8 May 1942) is a Swedish logician, philosopher, and mathematical statistician. He is internationally renowned for his work on the foundations of probability, statistics, mathematical logic, and computer scie ...
'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 professor at the University of North Carolina at Charlotte specialized in algorithmic complexity and cryptography. He is the inventor of IEEE P1363 cryptographic standards SRP5 and WANG-KE and has contri ...
that these randomness notions are generally different.
Randomness occurs in numbers such as
log(2) and
pi. 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 Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
:
In statistics
In statistics, randomness is commonly used to create
simple random samples
In statistics, a simple random sample (or SRS) is a subset of individuals (a sample) chosen from a larger set (a population) in which a subset of individuals are chosen randomly, all with the same probability. It is a process of selecting a sam ...
. 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
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 a ...
, 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 Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
and
Alfréd Rényi
Alfréd Rényi (20 March 1921 – 1 February 1970) was a Hungarian mathematician known for his work in probability theory, though he also made contributions in combinatorics, graph theory, and number theory.
Life
Rényi was born in Budapest to ...
, 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 hypothesis The random walk hypothesis is a financial theory stating that stock market prices evolve according to a random walk (so price changes are random) and thus cannot be predicted.
History
The concept can be traced to French broker Jules Regnault who pu ...
considers that asset prices in an organized
market
Market is a term used to describe concepts such as:
*Market (economics), system in which parties engage in transactions according to supply and demand
*Market economy
*Marketplace, a physical marketplace or public market
Geography
*Märket, an ...
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 is one of the oldest named cities in the world, having been continuously inhabited for perhaps 5,000 years. Situated in southern Europe, Athens became the leading city of Ancient Greece in the first millennium BC, and its cultural achieve ...
were chosen by lot instead of modern voting.
Randomness and religion
Randomness can be seen as conflicting with the
deterministic
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
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 characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...
, which states that
non-random
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ran ...
selection is applied to the results of random genetic variation.
Hindu
Hindus (; ) are people who religiously adhere to Hinduism.Jeffery D. Long (2007), A Vision for Hinduism, IB Tauris, , pages 35–37 Historically, the term has also been used as a geographical, cultural, and later religious identifier for ...
and
Buddhist
Buddhism ( , ), also known as Buddha Dharma and Dharmavinaya (), is an Indian religion or philosophical tradition based on teachings attributed to the Buddha. It originated in northern India as a -movement in the 5th century BCE, and ...
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) in Sanskrit means an action, work, or deed, and its effect or consequences. In Indian religions, the term more specifically refers to a principle of cause and effect, often descriptivel ...
. 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 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 a deity.
In classical civ ...
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 city-state (known as a polis) of Athens, comprising the city of Athens and the surrounding territory of Attica. Although Athens is the most famous ancient Greek democratic city- ...
was based on the concept of
isonomia
''Isonomia'' (ἰσονομία "equality of political rights,"Henry George Liddell, Robert Scott, A Greek-English LexiconThe Athenian Democracy in the Age of Demosthenes", Mogens Herman Hansen, , p. 81-84 from the Greek ἴσος ''isos'', ...
(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
randomization Randomization is the process of making something random. Randomization is not haphazard; instead, a random process is a sequence of random variables describing a process whose outcomes do not follow a deterministic pattern, but follow an evolution d ...
, such as selecting
juror
A jury is a sworn body of people (jurors) convened to hear evidence and 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 o ...
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 ...
lotteries.
Games: Random numbers were first investigated in the context of
gambling
Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three el ...
, 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 values, commonly as part of tabletop games, including dice games, board games, role-playing g ...
,
shuffling playing cards
Shuffling is a procedure used to randomize a deck of playing cards to provide an element of chance in card games. Shuffling is often followed by a cut, to help ensure that the shuffler has not manipulated the outcome.
__TOC__
Techniques
Over ...
, and
roulette
Roulette is a casino game named after the French word meaning ''little wheel'' which was likely developed from the Italian game Biribi''.'' In the game, a player may choose to place a bet on a single number, various groupings of numbers, the ...
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 Board
A gaming control board (GCB), also called by various names including gambling control board, casino control board, gambling board, and gaming commission, is a government agency charged with regulating casino and other types of gaming in a defined ...
s. Random drawings are also used to determine
lottery
A lottery is a form of gambling that involves the drawing of numbers at random for a prize. Some governments outlaw lotteries, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of ...
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 straws Drawing straws is a selection method, or a form of sortition, that is used by a group to choose one member of the group to perform a task after none has volunteered for it. The same practice can be used also to choose one of several volunteers, shou ...
).
Sports: Some sports, including
American football
American football (referred to simply as football in the United States and Canada), also known as gridiron, is a team sport played by two teams of eleven players on a rectangular field with goalposts at each end. The offense, the team with ...
, use
coin toss
A coin is a small, flat (usually depending on the country or value), round piece of metal or plastic used primarily as a medium of exchange or legal tender. They are standardized in weight, and produced in large quantities at a mint in order to ...
es to randomly select starting conditions for games or
seed
A seed is an embryonic plant enclosed in a protective outer covering, along with a food reserve. The formation of the seed is a part of the process of reproduction in seed plants, the spermatophytes, including the gymnosperm and angiospe ...
tied teams for
postseason play. The
National Basketball Association
The National Basketball Association (NBA) is a professional basketball 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 major professional sports leagues in the United S ...
uses a weighted
lottery
A lottery is a form of gambling that involves the drawing of numbers at random for a prize. Some governments outlaw lotteries, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of ...
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 survey or a poll (although strictly a poll is an actual election) is a human research survey of public opinion from a particular sample. Opinion polls are usually designed to represent the opinions ...
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 defines quality control as "a part of quality management focused on fulfilling quality requirements".
This approach places ...
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 determi ...
and in
genetic algorithm
In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to gene ...
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 form of scientific experiment used to control factors not under direct experimental control. Examples of RCTs are clinical trials that compare the effects of drugs, surgical te ...
).
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 occultic, standardized process or ritual. Used in various forms throughout histor ...
such as
cleromancy
Cleromancy is a form of sortition (casting of 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 a deity.
In classical civ ...
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 unimpeded.
Free will is closely linked to the concepts of moral responsibility, praise, culpability, sin, and other judgements which apply only to actio ...
and
Determinism
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
for more).
Generation
![Roulette wheel](https://upload.wikimedia.org/wikipedia/commons/2/26/Roulette_wheel.jpg)
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 particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, but also
hardware random number generator
In computing, a hardware random number generator (HRNG) or true random number generator (TRNG) is a device that generates random numbers from a physical process, rather than by means of an algorithm. Such devices are often based on microscopic ...
s).
# ''Randomness'' coming from the initial conditions. This aspect is studied by
chaos theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
, and is observed in systems whose behavior is very sensitive to small variations in initial conditions (such as
pachinko
is a mechanical game originating in Japan that is used as an arcade game, and much more frequently for gambling. Pachinko fills a niche in Japanese gambling comparable to that of the slot machine in the West as a form of low-stakes, low-st ...
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 values, commonly as part of tabletop games, including dice games, board games, role-playing g ...
).
# ''Randomness'' intrinsically generated by the system. This is also called
pseudorandomness
A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process.
Background
The generation of random numbers has many uses, such as for random ...
, and is the kind used in
pseudo-random number generator
A pseudorandom number generator (PRNG), also known as a deterministic random bit generator (DRBG), is an algorithm for generating a sequence of numbers whose properties approximate the properties of sequences of random numbers. The PRNG-generate ...
s. There are many algorithms (based on
arithmetics
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ce ...
or
cellular automaton
A cellular automaton (pl. cellular automata, abbrev. CA) is a discrete model of computation studied in automata theory. Cellular automata are also called cellular spaces, tessellation automata, homogeneous structures, cellular structures, tessel ...
) 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 has many uses in science, art, statistics, cryptography, gaming, gambling, and other fields. For example, random assignment in randomized controlled trials helps scientists to test hypotheses, and random numbers or pseudorandom number ...
have led to many different methods for generating random data. These methods may vary as to how unpredictable or
statistically random 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 numbers or symbols that cannot be reasonably predicted better than by random chance is generated. This means that the particular out ...
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 for tasks such as selected randomness, random samples. This was much more effective than manually selecting the random samples (with dice, cards, etc.). Nowadays, tables of random numbers have been ...
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, discrete transforms are mathematical transforms, often linear transforms, of signals between discrete domains, such as between discrete time and discrete frequency.
Many common integral transforms used in signal processing ...
s,
complexity
Complexity characterises the behaviour of a system or model whose components interaction, interact in multiple ways and follow local rules, leading to nonlinearity, randomness, collective dynamics, hierarchy, and emergence.
The term is generall ...
, 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 statistics of a multipartite quantum system do not admit an interpretation in terms of a local realistic theory. Quantum nonlocality has been experimen ...
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 fi ...
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 that involves the drawing of numbers at random for a prize. Some governments outlaw lotteries, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of ...
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
("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ex ...
'', 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.
![Monty open door](https://upload.wikimedia.org/wikipedia/commons/3/3f/Monty_open_door.svg)
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 mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
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
The Boy or Girl paradox surrounds a set of questions in probability theory, which are also known as The Two Child Problem, Mr. Smith's Children and the Mrs. Smith Problem. The initial formulation of the question dates back to at least 1959, when&nb ...
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) ...
, a game show scenario in which a car is hidden behind one of three doors, and two goats are hidden as
booby prize
A booby prize is a joke prize usually given in recognition of a terrible performance or last-place finish. A person who finishes last, for example, may receive a booby prize such as a worthless coin. Booby prizes are sometimes jokingly coveted a ...
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 or aleatorism, the noun associated with the adjectival aleatory and aleatoric, is a term popularised by the musical composer Pierre Boulez, but also Witold Lutosławski and Franco Evangelisti, for compositions resulting from "actio ...
*
Chaitin's constant
In the computer science subfield of algorithmic information theory, a Chaitin constant (Chaitin omega number) or halting probability is a real number that, informally speaking, represents the probability that a randomly constructed program will ...
*
Chance (disambiguation)
Chance may refer to:
Mathematics and Science
* In mathematics, likelihood of something (by way of the Likelihood function and/or Probability density function).
* Chance (statistics magazine), ''Chance'' (statistics magazine)
Places
* Chance, ...
*
Frequency probability
Frequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of a sequence, limit of its relative Frequency_(statistics), frequency in many trials (the long-run probability). Probabi ...
*
Indeterminism
Indeterminism is the idea that events (or certain events, or events of certain types) are not caused, or do not cause deterministically.
It is the opposite of determinism and related to chance. It is highly relevant to the philosophical probl ...
*
Nonlinear system
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
*
Probability interpretations
The word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly one be ...
*
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 expressing it through a set o ...
*
Pseudorandomness
A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process.
Background
The generation of random numbers has many uses, such as for random ...
*
Random.org—generates random numbers using atmospheric noise
*
Sortition
In governance, sortition (also known as selection by lottery, selection by lot, allotment, demarchy, stochocracy, aleatoric democracy, democratic lottery, and lottocracy) is the selection of political officials as a random sample from a larger ...
Notes
References
Further reading
* ''Randomness'' by
Deborah J. Bennett. Harvard University Press, 1998. .
* ''Random Measures, 4th ed.'' by
Olav Kallenberg
Olav Kallenberg (born 1939) is a probability theorist known for his work on exchangeable stochastic processes and for his graduate-level textbooks and monographs. Kallenberg is a professor of mathematics at Auburn University in Alabama in the US ...
. Academic Press, New York, London; Akademie-Verlag, Berlin, 1986. .
* ''The Art of Computer Programming. Vol. 2: Seminumerical Algorithms, 3rd ed.'' by
Donald E. Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist, mathematician, and professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer sc ...
. Reading, MA: Addison-Wesley, 1997. .
* ''
Fooled by Randomness
''Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets'' is a book by Nassim Nicholas Taleb that deals with the fallibility of human knowledge. It was first published in 2001. Updated editions were released a few years later. ...
, 2nd ed.'' by
Nassim Nicholas Taleb
Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist whose work concerns problems of randomness, ...
. Thomson Texere, 2004. .
* ''Exploring Randomness'' by
Gregory Chaitin
Gregory John Chaitin ( ; born 25 June 1947) is an Argentine-American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer-t ...
. 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
Leonard Mlodinow (; November 26, 1954) is an American theoretical physicist and mathematician, screenwriter and author. In physics, he is known for his work on the large N expansion, a method of approximating the spectrum of atoms based on the ...
. Pantheon Books, New York, 2008. .
External links
QuantumLabQuantum random number generator with single photons as interactive experiment.
HotBitsgenerates random numbers from radioactive decay.
QRBGQuantum Random Bit Generator
QRNGFast 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
The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. Eac ...
{{Authority control
Cryptography
Statistical randomness