Randomness

# Randomness

Overview
Randomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability
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:...

(or lack thereof) of events.

The Oxford English Dictionary
Oxford English Dictionary
The Oxford English Dictionary , published by the Oxford University Press, is the self-styled premier dictionary of the English language. Two fully bound print editions of the OED have been published under its current name, in 1928 and 1989. The first edition was published in twelve volumes , and...

defines 'random' as "Having no definite aim or purpose; not sent or guided in a particular direction; made, done, occurring, etc., without method or conscious choice; haphazard." This concept of randomness suggests a non-order or non-coherence in a sequence of symbol
Symbol
A symbol is something which represents an idea, a physical entity or a process but is distinct from it. The purpose of a symbol is to communicate meaning. For example, a red octagon may be a symbol for "STOP". On a map, a picture of a tent might represent a campsite. Numerals are symbols for...

s or steps
Procedure (term)
A procedure is a sequence of actions or operations which have to be executed in the same manner in order to always obtain the same result under the same circumstances ....

, such that there is no intelligible pattern or combination.

Applied usage in science, mathematics and statistics recognizes a lack of predictability
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:...

when referring to randomness, but admits regularities in the occurrences of events whose outcomes are not certain.
Discussion

Quotations

For I do not believe that it is through the interference of Divine Providence ... that the spittle of a certain person moved, fell on a certain gnat in a certain place, and killed it.

Maimonides, Guide for the Perplexed|Guide for the Perplexed, 12th century, Friedländer translation

The generation of random numbers is too important to be left to chance.

Robert R. Coveyou, Oak Ridge National Laboratory, 1969

How dare we speak of the laws of chance? Is not chance the antithesis of all law?

Joseph Bertrand, Calcul des probabilités, 1889

Random numbers should not be generated with a method chosen at random.

Donald Knuth|Donald E. Knuth, The Art of Computer Programming|The Art of Computer Programming, Vol. II, 1969, section 3.1

The sun comes up just about as often as it goes down, in the long run, but this doesn't make its motion random.

Donald Knuth|Donald E. Knuth, The Art of Computer Programming|The Art of Computer Programming, Vol. II, 1969, section 3.3.2
Encyclopedia
Randomness has somewhat differing meanings as used in various fields. It also has common meanings which are connected to the notion of predictability
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:...

(or lack thereof) of events.

The Oxford English Dictionary
Oxford English Dictionary
The Oxford English Dictionary , published by the Oxford University Press, is the self-styled premier dictionary of the English language. Two fully bound print editions of the OED have been published under its current name, in 1928 and 1989. The first edition was published in twelve volumes , and...

defines 'random' as "Having no definite aim or purpose; not sent or guided in a particular direction; made, done, occurring, etc., without method or conscious choice; haphazard." This concept of randomness suggests a non-order or non-coherence in a sequence of symbol
Symbol
A symbol is something which represents an idea, a physical entity or a process but is distinct from it. The purpose of a symbol is to communicate meaning. For example, a red octagon may be a symbol for "STOP". On a map, a picture of a tent might represent a campsite. Numerals are symbols for...

s or steps
Procedure (term)
A procedure is a sequence of actions or operations which have to be executed in the same manner in order to always obtain the same result under the same circumstances ....

, such that there is no intelligible pattern or combination.

Applied usage in science, mathematics and statistics recognizes a lack of predictability
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:...

when referring to randomness, but admits regularities in the occurrences of events whose outcomes are not certain. For example, when throwing 2 dice and counting the total, we can say 7 will randomly occur twice as often as 4. This view, where randomness simply refers to situations in which the certainty of the outcome is at issue, is the one taken when referring to concepts of chance, probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, and information entropy
Information entropy
In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits...

. In these situations randomness implies a measure of uncertainty and notions of haphazardness are irrelevant.

The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events. A random process is a sequence of random variables describing a process whose outcomes do not follow a deterministic
Determinism
Determinism is the general philosophical thesis that states that for everything that happens there are conditions such that, given them, nothing else could happen. There are many versions of this thesis. Each of them rests upon various alleged connections, and interdependencies of things and...

pattern, but follow an evolution described by probability distributions
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

. These and other constructs are extremely useful in the probability calculus.

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

to signify well-defined statistical properties, such as a lack of bias
Bias (statistics)
A statistic is biased if it is calculated in such a way that it is systematically different from the population parameter of interest. The following lists some types of, or aspects of, bias which should not be considered mutually exclusive:...

or correlation
Correlation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

. Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

s, which rely on random input, are important techniques in science, as, for instance, in computational science.

Random selection is a method of selecting items (oftentimes called units) from a population where the probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

of choosing a specific item is the proportion of those items in the population. For example, if we have a bowl of 100 marbles with 10 red (and any red marble is indistinguishable from any other red marble) and 90 blue (and any blue marble is indistinguishable from any other blue marble), 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 the population consists of items that are all distinguishable, a random selection mechanism would require equal probabilities for any item to be chosen. That is, if the section process is such that each member of a population, of say research subjects, has the same probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

of being chosen then we can say the selection process is random. Random selection can be an official method to resolve tied
Tie (draw)
To tie or draw is to finish a competition with identical or inconclusive results. The word "tie" is usually used in North America for sports such as American football. "Draw" is usually used in the United Kingdom, Ireland and the Commonwealth of Nations and it is usually used for sports such as...

elections in some jurisdictions and is even an ancient method of divination
Divination
Divination is the attempt to gain insight into a question or situation by way of an occultic standardized process or ritual...

, as in tarot
Tarot
The tarot |trionfi]] and later as tarocchi, tarock, and others) is a pack of cards , used from the mid-15th century in various parts of Europe to play a group of card games such as Italian tarocchini and French tarot...

, the I Ching
I Ching
The I Ching or "Yì Jīng" , also known as the Classic of Changes, Book of Changes and Zhouyi, is one of the oldest of the Chinese classic texts...

, and bibliomancy
Bibliomancy
Bibliomancy is the use of books in divination. The method of employing sacred books for 'magical medicine', for removing negative entities, or for divination is widespread in many religions of the world:-Terminology:...

. Its use in politics is very old, as office holders in Ancient Athens were chosen by lot, there being no voting.

## History

In ancient history, the concepts of chance and randomness were intertwined with that of fate. Many ancient peoples threw dice to determine fate, and this later evolved into games of chance. Most ancient cultures used various methods of divination
Divination
Divination is the attempt to gain insight into a question or situation by way of an occultic standardized process or ritual...

to attempt to circumvent randomness and fate.

The Chinese were perhaps the earliest people to formalize odds and chance 3,000 years ago. The Greek philosophers discussed randomness at length, but only in non-quantitative forms. It was only in the sixteenth century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of the calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...

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
Donald A. Venn FRS , was a British logician and philosopher. He is famous for introducing the Venn diagram, which is used in many fields, including set theory, probability, logic, statistics, and computer science....

wrote a chapter on "The conception of randomness" which included his view of the randomness of the digits of the number Pi
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

by using them to construct a random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...

in two dimensions.

The early part of the twentieth 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-twentieth century, ideas of algorithmic information theory
Algorithmic information theory
Algorithmic information theory is a subfield of information theory and computer science that concerns itself with the relationship between computation and information...

introduced new dimensions to the field via the concept of algorithmic randomness.

Although randomness had often been viewed as an obstacle and a nuisance for many centuries, in the twentieth 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 outperform the best deterministic methods.

## Randomness in science

Many scientific fields are concerned with randomness:
• Algorithmic probability
Algorithmic probability
In algorithmic information theory, algorithmic probability is a method of assigning a probability to each hypothesis that explains a given observation, with the simplest hypothesis having the highest probability and the increasingly complex hypotheses receiving increasingly small probabilities...

• Chaos theory
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

• Cryptography
Cryptography
Cryptography is the practice and study of techniques for secure communication in the presence of third parties...

• Game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

• Information theory
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...

• Pattern recognition
Pattern recognition
In machine learning, pattern recognition is the assignment of some sort of output value to a given input value , according to some specific algorithm. An example of pattern recognition is classification, which attempts to assign each input value to one of a given set of classes...

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

• Quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

• Statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

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

### In the physical sciences

In the 19th century, scientists used the idea of random motions of molecules in the development of statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

in order to explain phenomena in thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

and the properties of gases
Gas laws
The early gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between the pressure, volume and temperature of a sample of gas could be obtained which would hold for all gases...

.

According to several standard interpretations of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, microscopic phenomena are objectively random. That is, in an experiment where all causally relevant parameters are controlled, there will still be some aspects of the outcome which vary randomly. An example of such an experiment is placing a single unstable atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

in a controlled environment; it cannot be predicted how long it will take for the atom to decay; only the probability of decay within a given time can be calculated. Thus, quantum mechanics does not specify the outcome of individual experiments but only the probabilities. Hidden variable theories
Hidden variable theory
Historically, in physics, hidden variable theories were espoused by some physicists who argued that quantum mechanics is incomplete. These theories argue against the orthodox interpretation of quantum mechanics, which is the Copenhagen Interpretation...

are inconsistent with the view that nature contains irreducible randomness: such theories posit that in the processes that appear random, properties with a certain statistical distribution are somehow at work "behind the scenes" determining the outcome in each case.

### In biology

The modern evolutionary synthesis
Modern evolutionary synthesis
The modern evolutionary synthesis is a union of ideas from several biological specialties which provides a widely accepted account of evolution...

ascribes the observed diversity of life to natural selection
Natural selection
Natural selection is the nonrandom process by which biologic traits become either more or less common in a population as a function of differential reproduction of their bearers. It is a key mechanism of evolution....

, in which some random genetic mutation
Mutation
In molecular biology and genetics, mutations are changes in a genomic sequence: the DNA sequence of a cell's genome or the DNA or RNA sequence of a virus. They can be defined as sudden and spontaneous changes in the cell. Mutations are caused by radiation, viruses, transposons and mutagenic...

s are retained in the gene pool
Gene pool
In population genetics, a gene pool is the complete set of unique alleles in a species or population.- Description :A large gene pool indicates extensive genetic diversity, which is associated with robust populations that can survive bouts of intense selection...

due to the non-random improved chance for survival and reproduction that those mutated genes confer on individuals who possess them.

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 that appear on a person's skin is controlled by genes and exposure to light; whereas the exact location of individual freckles seems to be random.

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
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

arose from attempts to formulate mathematical descriptions of chance events, originally in the context of gambling
Gambling
Gambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods...

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

is used to infer the underlying probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

of a collection of empirical observations. For the purposes of simulation
Simulation
Simulation is the imitation of some real thing available, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviours of a selected physical or abstract system....

, it is necessary to have a large supply of random numbers
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 X1,...,Xn be independent random variables...". Yet as D. H. Lehmer stated in...

or means to generate them on demand.

Algorithmic information theory
Algorithmic information theory
Algorithmic information theory is a subfield of information theory and computer science that concerns itself with the relationship between computation and information...

studies, among other topics, what constitutes a random sequence
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 X1,...,Xn be independent random variables...". Yet as D. H. Lehmer stated in...

. The central idea is that a string of bit
Bit
A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...

s is random if and only if it is shorter than any computer program that can produce that string (Kolmogorov randomness)—this means that random strings are those that cannot be compressed
Data compression
In computer science and information theory, data compression, source coding or bit-rate reduction is the process of encoding information using fewer bits than the original representation would use....

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

and his student Per Martin-Löf
Per Martin-Löf
Per Erik Rutger Martin-Löf 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 science. Since the late 1970s, Martin-Löf's publications have been mainly in...

, Ray Solomonoff
Ray Solomonoff
Ray Solomonoff was the inventor of algorithmic probability, and founder of algorithmic information theory, He was an originator of the branch of artificial intelligence based on machine learning, prediction and probability...

, and Gregory Chaitin
Gregory Chaitin
Gregory John Chaitin is an Argentine-American mathematician and computer scientist.-Mathematics and computer science:Beginning in 2009 Chaitin has worked on metabiology, a field parallel to biology dealing with the random evolution of artificial software instead of natural software .Beginning in...

.

In mathematics, there must be an infinite expansion of information for randomness to exist. This can best be seen with an example. Given a random sequence of three-bit numbers, each number can have one of only eight possible values:

000, 001, 010, 011, 100, 101, 110, 111

Therefore, as the random sequence progresses, it must recycle the values it previously used. In order to increase the information space, another bit may be added to each possible number, giving 16 possible values from which to pick a random number. It could be said that the random four-bit number sequence is more random than the three-bit one. This suggests that in order to have true randomness, there must be an infinite expansion of the information space.

Randomness is said to occur in numbers such as log (2)
Binary logarithm
In mathematics, the binary logarithm is the logarithm to the base 2. It is the inverse function of n ↦ 2n. The binary logarithm of n is the power to which the number 2 must be raised to obtain the value n. This makes the binary logarithm useful for anything involving powers of 2,...

and Pi
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

. The decimal digits of Pi constitute an infinite sequence and "never repeat in a cyclical fashion". Numbers like pi are also thought to be normal
Normal number
In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2,...

, which means that their digits are random in a certain statistical sense.

Pi certainly seems to behave this way. In the first six billion decimal places of pi, each of the digits from 0 through 9 shows up about six hundred million times. Yet such results, conceivably accidental, do not prove normality even in base 10, much less normality in other number bases.

### In statistics

In statistics, randomness is commonly used to create simple random samples. This allows surveys to be done with completely random groups of people to allow realistic data. Common methods of doing this are "drawing names out of a hat" or using a random digit chart. A random digit chart is simply a large table of random digits.

### In information science

In information science, irrelevant or meaningless data is considered to be noise. Noise consists of a large number of transient disturbances with a statistically randomized time distribution.

In communication theory
Communication theory
Communication theory is a field of information and mathematics that studies the technical process of information and the human process of human communication.- History :- Origins :...

, 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 Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...

and Alfréd Rényi
Alfréd Rényi
Alfréd Rényi was a Hungarian mathematician who made contributions in combinatorics, graph theory, number theory but mostly in probability theory.-Life:...

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
Random walk hypothesis
The random walk hypothesis is a financial theory stating that stock market prices evolve according to a random walk and thus the prices of the stock market cannot be predicted. It is consistent with the efficient-market hypothesis....

considers that asset prices in an organized market
Market
A market is one of many varieties of systems, institutions, procedures, social relations and infrastructures whereby parties engage in exchange. While parties may exchange goods and services by barter, most markets rely on sellers offering their goods or services in exchange for money from buyers...

evolve at random.

Other so-called random factors intervene in trends and patterns to do with supply-and-demand distributions. As well as this, the random factor of the environment itself results in fluctuations in stock and broker markets.

### Randomness versus unpredictability

Randomness, as opposed to unpredictability, is held to be an objective property - determinists believe it is an objective fact that randomness does not in fact exist. Also, what appears random to one observer may not appear random to another. Consider two observers of a sequence of bits, when only one of whom has the cryptographic key needed to turn the sequence of bits into a readable message. For that observer the message is not random, but it is unpredictable for the other.

One of the intriguing aspects of random processes is that it is hard to know whether a process is truly random. An observer may suspect that there is some "key" that unlocks the message. This is one of the foundations of superstition
Superstition
Superstition is a belief in supernatural causality: that one event leads to the cause of another without any process in the physical world linking the two events....

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

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

.

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

, there is no randomness in the universe, only unpredictability
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:...

, since there is only one possible outcome to all events in the universe. A follower of the narrow frequency interpretation of probability could assert that no event can be said to have probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

, since there is only one universal outcome. On the other hand, under the rival Bayesian interpretation of probability
Bayesian probability
Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...

there is no objection to the use of probabilities in order to represent a lack of complete knowledge of the outcomes.

Some mathematically defined sequences, such as the decimals of pi
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

mentioned above, exhibit some of the same characteristics as random sequences, but because they are generated by a describable mechanism, they are called pseudorandom. To an observer who does not know the mechanism, a pseudorandom sequence is unpredictable.

Chaotic systems
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

are unpredictable in practice due to their extreme sensitivity to initial conditions. Whether or not they are unpredictable in terms of computability theory is a subject of current research. At least in some disciplines of computability theory, the notion of randomness is identified with computational unpredictability.

Individual events that are random may still be precisely described en masse, usually in terms of probability or expected value. For instance, quantum mechanics allows a very precise calculation of the half-lives of atoms even though the process of atomic decay is random. More simply, although a single toss of a fair coin cannot be predicted, its general behavior can be described by saying that if a large number of tosses are made, roughly half of them will show up heads. Ohm's law
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...

and the kinetic theory of gases
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...

are non-random macroscopic
Macroscopic
The macroscopic scale is the length scale on which objects or processes are of a size which is measurable and observable by the naked eye.When applied to phenomena and abstract objects, the macroscopic scale describes existence in the world as we perceive it, often in contrast to experiences or...

phenomena that are assumed to be random at the microscopic
Microscope
A microscope is an instrument used to see objects that are too small for the naked eye. The science of investigating small objects using such an instrument is called microscopy...

level.

## Randomness and religion

Some theologians have attempted to resolve the apparent contradiction between an omniscient deity, or a first cause, and free will
Free will
"To make my own decisions whether I am successful or not due to uncontrollable forces" -Troy MorrisonA pragmatic definition of free willFree will is the ability of agents to make choices free from certain kinds of constraints. The existence of free will and its exact nature and definition have long...

using randomness. Discordians
Discordianism
Discordianism is a religion based on the worship of Eris , the Greco-Roman goddess of strife. It was founded circa 1958–1959 after the publication of its holy book the Principia Discordia, written by Malaclypse the Younger and Omar Khayyam Ravenhurst after a series of shared hallucinations at a...

have a strong belief in randomness and unpredictability. Hindu
Hindu
Hindu refers to an identity associated with the philosophical, religious and cultural systems that are indigenous to the Indian subcontinent. As used in the Constitution of India, the word "Hindu" is also attributed to all persons professing any Indian religion...

and Buddhist philosophies state that any event is the result of previous events (karma
Karma
Karma in Indian religions is the concept of "action" or "deed", understood as that which causes the entire cycle of cause and effect originating in ancient India and treated in Hindu, Jain, Buddhist and Sikh philosophies....

), and as such, there is no such thing as a random event or a first event.

Martin Luther
Martin Luther
Martin Luther was a German priest, professor of theology and iconic figure of the Protestant Reformation. He strongly disputed the claim that freedom from God's punishment for sin could be purchased with money. He confronted indulgence salesman Johann Tetzel with his Ninety-Five Theses in 1517...

, the forefather of Protestantism
Protestantism
Protestantism is one of the three major groupings within Christianity. It is a movement that began in Germany in the early 16th century as a reaction against medieval Roman Catholic doctrines and practices, especially in regards to salvation, justification, and ecclesiology.The doctrines of the...

, believed that there was nothing random based on his understanding of the Bible
Bible
The Bible refers to any one of the collections of the primary religious texts of Judaism and Christianity. There is no common version of the Bible, as the individual books , their contents and their order vary among denominations...

. As an outcome of his understanding of randomness, he strongly felt that free will was limited to low-level decision making by humans. Therefore, when someone sins against another, decision making is only limited to how one responds, preferably through forgiveness and loving actions. He believed, based on Biblical scripture, that humans cannot will themselves faith, salvation, sanctification, or other gifts from God. Additionally, the best people could do, according to his understanding, was not sin, but they fall short, and free will cannot achieve this objective. Thus, in his view, absolute free will and unbounded randomness are severely limited to the point that behaviors may even be patterned or ordered and not random. This is a point emphasized by the field of behavioral psychology.

These notions and more in Christianity often lend to a highly deterministic worldview and that the concept of random events is not possible. Especially, if purpose is part of this universe, then randomness, by definition, is not possible. This is also one of the rationales for religious opposition to evolution
Evolution
Evolution is any change across successive generations in the heritable characteristics of biological populations. Evolutionary processes give rise to diversity at every level of biological organisation, including species, individual organisms and molecules such as DNA and proteins.Life on Earth...

, where, according to theory, (non-random) selection is applied to the results of random genetic variation.

Donald Knuth
Donald Knuth
Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...

, a Stanford computer scientist and Christian commentator, remarks that he finds pseudorandom numbers useful and applies them with purpose. He then extends this thought to God who may use randomness with purpose to allow free will to certain degrees. Knuth believes that God is interested in people's decisions and limited free will allows a certain degree of decision making. Knuth, based on his understanding of quantum computing and entanglement, comments that God exerts dynamic control over the world without violating any laws of physics, suggesting that what appears to be random to humans may not, in fact, be so random.

C. S. Lewis
C. S. Lewis
Clive Staples Lewis , commonly referred to as C. S. Lewis and known to his friends and family as "Jack", was a novelist, academic, medievalist, literary critic, essayist, lay theologian and Christian apologist from Belfast, Ireland...

, a 20th-century Christian philosopher, discussed free will at length. On the matter of human will, Lewis wrote: "God willed the free will of men and angels in spite of His knowledge that it could lead in some cases to sin and thence to suffering: i.e., He thought freedom worth creating even at that price." In his radio broadcast, Lewis indicated that God "gave [humans] free will. He gave them free will because a world of mere automata could never love..."

In some contexts, procedures that are commonly perceived as randomizers—drawing lots or the like —are used for divination, e.g., to reveal the will of the gods; see e.g. Cleromancy
Cleromancy
Cleromancy is a form of divination using sortition, casting of lots, or casting bones or stones, in which an outcome is determined by means that normally would be considered random, such as the rolling of dice, but are sometimes believed to reveal the will of God, or other supernatural entities.-In...

.

## Applications and use of randomness

In most of its mathematical, political, social and religious use, randomness is used for its innate "fairness" and lack of bias.

Political: Athenian democracy
Athenian democracy
Athenian democracy developed in the Greek city-state of Athens, comprising the central city-state of Athens and the surrounding territory of Attica, around 508 BC. Athens is one of the first known democracies. Other Greek cities set up democracies, and even though most followed an Athenian model,...

was based on the concept of isonomia
Isonomia
Isonomia was a word used by Ancient Greek writers such as Herodotus and Thucydides to refer to some kind of popular government...

(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
Sortition
In politics, sortition is the selection of decision makers by lottery. The decision-makers are chosen as a random sample from a larger pool of candidates....

is now restricted to selecting jurors in Anglo-Saxon legal systems and in situations where "fairness" is approximated by randomization
Randomization
Randomization is the process of making something random; this means:* Generating a random permutation of a sequence .* Selecting a random sample of a population ....

, such as selecting jurors and military draft
Conscription
Conscription is the compulsory enlistment of people in some sort of national service, most often military service. Conscription dates back to antiquity and continues in some countries to the present day under various names...

lotteries.

Social: Random numbers were first investigated in the context of gambling
Gambling
Gambling is the wagering of money or something of material value on an event with an uncertain outcome with the primary intent of winning additional money and/or material goods...

, and many randomizing devices, such as dice
Dice
A die is a small throwable object with multiple resting positions, used for generating random numbers...

, shuffling playing cards, and roulette
Roulette
Roulette is a casino game named after a French diminutive for little wheel. In the game, players may choose to place bets on either a single number or a range of numbers, the colors red or black, or whether the number is odd or even....

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
Gaming Control Board
A gaming control board , 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 geographical area, usually a state, and of enforcing...

s. Random drawings are also used to determine lottery
Lottery
A lottery is a form of gambling which involves the drawing of lots for a prize.Lottery is outlawed by some governments, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of regulation of lottery by governments...

winners. Throughout history, randomness has been used for games of chance and to select out individuals for an unwanted task in a fair way (see drawing straws
Drawing straws
Drawing straws is a selection method that is used by a group to choose one member of the group to perform a task after none has volunteered for it...

).

Sports: Some sports, including American Football
American football
American football is a sport played between two teams of eleven with the objective of scoring points by advancing the ball into the opposing team's end zone. Known in the United States simply as football, it may also be referred to informally as gridiron football. The ball can be advanced by...

, use coin tosses to randomly select starting conditions for games or seed
Seed (sports)
A seed is a preliminary ranking that can be used in arranging a sports tournament. It is called a seed because of the analogy with plants where the seed might grow into a top rank at the end of that tournament, or might instead wither away...

tied teams for postseason play. The National Basketball Association
The National Basketball Association is the pre-eminent men's professional basketball league in North America. It consists of thirty franchised member clubs, of which twenty-nine are located in the United States and one in Canada...

uses a weighted lottery
NBA Draft Lottery
The NBA Draft Lottery is an annual event held by the National Basketball Association in which the teams who had missed the playoffs in the previous season, or teams who hold the draft rights of another team that missed the playoffs in the previous season, participate in a lottery process to...

to order teams in its draft.

Mathematical: Random numbers are also used where their use is mathematically important, such as sampling for opinion poll
Opinion poll
An opinion poll, sometimes simply referred to as a poll is a survey of public opinion from a particular sample. Opinion polls are usually designed to represent the opinions of a population by conducting a series of questions and then extrapolating generalities in ratio or within confidence...

s and for statistical sampling in quality control
Quality control
Quality control, or QC for short, is a process by which entities review the quality of all factors involved in production. This approach places an emphasis on three aspects:...

systems. Computational solutions for some types of problems use random numbers extensively, such as in the Monte Carlo method
Monte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...

and in genetic algorithm
Genetic algorithm
A genetic algorithm is a search heuristic that mimics the process of natural evolution. This heuristic is routinely used to generate useful solutions to optimization and search problems...

s.

Medicine: Random allocation of a clinical intervention is used to reduce bias in controlled trials (e.g., randomized controlled trials).

Religious: Although not intended to be random, various forms of divination
Divination
Divination is the attempt to gain insight into a question or situation by way of an occultic standardized process or ritual...

such as cleromancy
Cleromancy
Cleromancy is a form of divination using sortition, casting of lots, or casting bones or stones, in which an outcome is determined by means that normally would be considered random, such as the rolling of dice, but are sometimes believed to reveal the will of God, or other supernatural entities.-In...

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
"To make my own decisions whether I am successful or not due to uncontrollable forces" -Troy MorrisonA pragmatic definition of free willFree will is the ability of agents to make choices free from certain kinds of constraints. The existence of free will and its exact nature and definition have long...

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

).

### Generating randomness

It is generally accepted that there exist three mechanisms responsible for (apparently) random behavior in systems:
1. Randomness coming from the environment (for example, Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

, but also hardware random number generator
Hardware random number generator
In computing, a hardware random number generator is an apparatus that generates random numbers from a physical process. Such devices are often based on microscopic phenomena that generate a low-level, statistically random "noise" signal, such as thermal noise or the photoelectric effect or other...

s)
2. Randomness coming from the initial conditions. This aspect is studied by chaos theory
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

and is observed in systems whose behavior is very sensitive to small variations in initial conditions (such as pachinko
Pachinko
is a type of game originating in Japan, and used as both a form of recreational arcade game and much more frequently as a gambling device, filling a niche in gambling in Japan comparable to that of the slot machine in Western gambling. A pachinko machine resembles a vertical pinball machine, but...

machines, dice
Dice
A die is a small throwable object with multiple resting positions, used for generating random numbers...

...).
3. Randomness intrinsically generated by the system. This is also called pseudorandomness
Pseudorandomness
A pseudorandom process is a process that appears to be random but is not. Pseudorandom sequences typically exhibit statistical randomness while being generated by an entirely deterministic causal process...

and is the kind used in pseudo-random number generators. There are many algorithms (based on arithmetics or cellular automaton
Cellular automaton
A cellular automaton is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. It consists of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off"...

) to generate pseudorandom numbers. The behavior of the system can be determined by knowing the seed state
Random seed
A random seed is a number used to initialize a pseudorandom number generator.The choice of a good random seed is crucial in the field of computer security...

and the algorithm used. These methods are often quicker than getting "true" randomness from the environment.

The many applications of randomness
Applications of randomness
Randomness has many uses in gambling, statistics, cryptography, art, etc.These uses have different randomness requirements, which leads to the use of different randomization methods...

have led to many different methods for generating random data. These methods may vary as to how unpredictable or statistically random
Statistical randomness
A 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 roll, or the digits of π exhibit statistical randomness....

they are, and how quickly they can generate random numbers.

Before the advent of computational random number generators, generating large amounts of sufficiently random numbers (important in statistics) required a lot of work. Results would sometimes be collected and distributed as random number table
Random number table
Random number tables have been used in statistics for tasks such as selected random samples. This was much more effective than manually selecting the random samples...

s.

### Randomness measures and tests

There are many practical measures of randomness for a binary sequence. These include measures based on frequency, discrete transform
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....

s, and complexity
Complexity
In general usage, complexity tends to be used to characterize something with many parts in intricate arrangement. The study of these complex linkages is the main goal of complex systems theory. In science there are at this time a number of approaches to characterizing complexity, many of which are...

, or a mixture of these. These include tests
Randomness tests
The issue of randomness is an important philosophical and theoretical question.Many random number generators in use today generate what are called "random sequences" but they are actually the result of prescribed algorithms and so they are called pseudo-random number generators.These generators do...

by Kak, Phillips, Yuen, Hopkins, Beth and Dai, Mund, and Marsaglia and Zaman.

## Misconceptions/logical fallacies

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

### A number is "due"

Coupon collector's problem
In probability theory, the coupon collector's problem describes the "collect all coupons and win" contests. It asks the following question: Suppose that there are n coupons, from which coupons are being collected with replacement...

This argument is that "in a random selection of numbers, since all numbers will 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
Playing card
A playing card is a piece of specially prepared heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic, marked with distinguishing motifs and used as one of a set for playing card games...

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
Lottery
A lottery is a form of gambling which involves the drawing of lots for a prize.Lottery is outlawed by some governments, while others endorse it to the extent of organizing a national or state lottery. It is common to find some degree of regulation of lottery by governments...

number selection schemes. Truly random processes such as these do not have memory, making it impossible for past outcomes to affect future outcomes.

### 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 to be likely to come up more often in the future. This logic is valid only if the randomisation is biased, for example with a loaded die. If the die is fair, then previous rolls give no indication of future events.

In nature, events rarely occur with perfectly equal frequency. So observing outcomes to determine which events are likely to have a higher probability, makes sense. It is fallacious to apply this logic to systems which are designed so that all outcomes are equally likely, such as shuffled cards, dice and roulette wheels.

### Odds are never dynamic

In the beginning of a scenario, one might calculate the odds of a certain event. The fact is, as soon as one gains more information about that situation, they may need to re-calculate the odds.

If we are told that a woman has two children, and one of them is a girl, what are the odds that the other child is also a girl? Considering this new child independently, one might expect the odds that the other child is female are 1/2 (50%). By using mathematician Gerolamo Cardano
Gerolamo Cardano
Gerolamo Cardano was an Italian Renaissance mathematician, physician, astrologer and gambler...

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

(illustrating all possible outcomes), we see that the odds are actually only 1/3 (33%). This is because, for starters, the possibility space illustrates 4 ways of having these two children: boy-boy, girl-boy, boy-girl, and girl-girl. But we were given more information. Once we are told that one of the children is a female, we use this new information to eliminate the boy-boy scenario. Thus the probability space reveals that there are still 3 ways to have two children where one is a female: boy-girl, girl-boy, girl-girl. Only 1/3 of these scenarios would have the other child also be a girl. Using a probability space, we are less likely to miss one of the possible scenarios, or to neglect the importance of new information.

This technique provides insights in other situations such as the Monty Hall problem, a game show scenario in which a car is hidden behind one of three doors, and two goats are hidden as booby prize
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 get a booby prize such as a worthless coin. Booby prizes are sometimes humorously and jokingly coveted as an object of pride.Booby prizes, however,...

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 host then asks the player whether they would like to keep the decision they made, or switch and select the other door. Intuitively, one might think the contestant is simply choosing between two doors with equal probability, and the opportunity provided by the host makes no difference. Probability spaces reveal that the contestant has received new information, and can increase their chances of winning by changing to the other door.

### Ignoring variance

Whether it is a career in poker, as a salesperson, or even searching for the right partner to marry, variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

and randomness play an important role. Variance sometimes prevents people from drawing causal relationships, even after they have performed multiple experiments. Put simply, in a popular game, some bad players are likely to have winning streaks and good players are likely to have losing streaks. This also explains why coincidences
Coincidence
A coincidence is an event notable for its occurring in conjunction with other conditions, e.g. another event. As such, a coincidence occurs when something uncanny, accidental and unexpected happens under conditions named, but not under a defined relationship...

should be considered skeptically; rare things, by definition, occasionally happen (e.g.
the sudden death of hundreds of animals
2010–2011 midwinter animal mass death events
The 2010–2011 midwinter animal mass death events have gained considerable publicity worldwide. Media attention was particularly high for this period. This is despite the fact that the mass deaths of fish and of birds are quite common...

).

## Books

• Randomness by Deborah J. Bennett. Harvard University Press, 1998. ISBN 0-674-10745-4.
• Random Measures, 4th ed. by Olav Kallenberg. Academic Press, New York, London; Akademie-Verlag, Berlin, 1986. MR0854102.
• The Art of Computer Programming. Vol. 2: Seminumerical Algorithms, 3rd ed. by Donald E. Knuth
Donald Knuth
Donald Ervin Knuth is a computer scientist and Professor Emeritus at Stanford University.He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms...

• Fooled by Randomness
Fooled by Randomness
Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets is a book written by Nassim Nicholas Taleb about the fallibility of human knowledge.-Reaction:The book was selected by Fortune as one of the 75 "Smartest Books of All Time."...

, 2nd ed. by Nassim Nicholas Taleb. Thomson Texere, 2004. ISBN 1-58799-190-X.
• Exploring Randomness by Gregory Chaitin
Gregory Chaitin
Gregory John Chaitin is an Argentine-American mathematician and computer scientist.-Mathematics and computer science:Beginning in 2009 Chaitin has worked on metabiology, a field parallel to biology dealing with the random evolution of artificial software instead of natural software .Beginning in...

. Springer-Verlag London, 2001. ISBN 1-85233-417-7.
• Random by Kenneth Chan includes a "Random Scale" for grading the level of randomness.

• Aleatory
Aleatory
Aleatoricism is the incorporation of chance into the process of creation, especially the creation of art or media. The word derives from the Latin word alea, the rolling of dice...

• Chaitin's constant
Chaitin's constant
In the computer science subfield of algorithmic information theory, a Chaitin constant or halting probability is a real number that informally represents the probability that a randomly constructed program will halt...

• Chance (disambiguation)
• Frequency probability
Frequency probability
Frequency probability is the interpretation of probability that defines an event's probability as the limit of its relative frequency in a large number of trials. The development of the frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the...

• Nonlinear system
• Probability interpretations
Probability interpretations
The word probability has been used in a variety of ways since it was first coined in relation to games of chance. Does probability measure the real, physical tendency of something to occur, or is it just a measure of how strongly one believes it will occur? In answering such questions, we...