Randomness has somewhat disparate meanings as used in several different fields. It also has common meanings which may have loose connections with some of those more definite meanings. The Oxford English Dictionary The Oxford English Dictionary , published by the Oxford University Press, is a dictionary of the English language. Two fully-bound print editions of the OED have been published under its current name, in 1928 and 1989. As of December 2008[update], the editors had completed one quarter of a third edition defines "random" thus:

Having no definite aim or purpose; not sent or guided in a particular direction; made, done, occurring, etc., without method or conscious choice; haphazard.

Also, in statistics, as:

Governed by or involving equal chances for each of the actual or hypothetical members of a population; (also) produced or obtained by such a process, and therefore unpredictable in detail.

Closely connected, therefore, with the concepts of chance, probability Probability is a way of expressing knowledge or belief that an event will occur or has occurred. The concept has been given an exact mathematical meaning in probability theory, which is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of, and information entropy In information theory, entropy is a measure of the uncertainty associated with a random variable. The term by itself in this context usually refers to the Shannon entropy, which quantifies, in the sense of an expected value, the information contained in a message, usually in units such as bits. Equivalently, the Shannon entropy is a measure of the, randomness implies a lack of predictability Predictability is the degree to which a correct prediction or forecast of a system's state can be made either qualitatively or quantitatively. More formally, in statistics, a random process In probability theory, a stochastic process, or sometimes random process, is the counterpart to a deterministic process . Instead of dealing with only one possible reality of how the process might evolve under time (as is the case, for example, for solutions of an ordinary differential equation), in a stochastic or random process there is some is a repeating process whose outcomes follow no describable deterministic Determinism is the philosophical view that every event, including human cognition, behaviour, decision, and action is causally determined (completely predictable) by previous events pattern, but follow a probability distribution In probability theory and statistics, a probability distribution identifies either the probability of each value of a random variable , or the probability of the value falling within a particular interval (when the variable is continuous). The probability distribution describes the range of possible values that a random variable can attain and the, such that the relative probability of the occurrence of each outcome can be approximated or calculated. For example, the rolling of a fair six-sided die in neutral conditions may be said to produce random results, because one cannot compute, before a roll, what number will show up. However, the probability of rolling any one of the six rollable numbers can be calculated, assuming that each is equally likely.

Randomness is a concept of non-order or non-coherence in a sequence of symbols A symbol is something such as an object, picture, written word, sound, or particular mark that represents something else by association, resemblance, or convention. For example, a red octagon may be a symbol for "STOP". On maps, crossed sabres may indicate a battlefield. Numerals are symbols for numbers . All language consists of symbols or steps, such that there is no intelligible pattern or combination.

The term is often used in statistics Statistics is the formal science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments to signify well-defined statistical properties, such as a lack of bias or correlation In statistics, correlation and dependence are any of a broad class of statistical relationships between two or more random variables or observed data values. Monte Carlo Methods 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 simulating physical and mathematical systems. Because of their reliance on repeated computation of random or pseudo-random numbers, these methods are most suited to calculation by a, which rely on random input, are important techniques in science, as, for instance, in computational science Computational science is the field of study concerned with constructing mathematical models and quantitative analysis techniques and using computers to analyse and solve scientific problems. In practical use, it is typically the application of computer simulation and other forms of computation to problems in various scientific disciplines.^[1] Random selection is an official method to resolve tied 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 and the Commonwealth of Nations and it is usually used for sports such as Football (soccer) and Australian rules elections in some jurisdictions^[2] and is even an ancient method of divination Divination is the attempt to gain insight into a question or situation by way of a standardized process or ritual. Diviners ascertain their interpretations of how a querent should proceed by reading signs, events, or omens, or through alleged contact with a supernatural agency. Divination can be seen as a systematic method with which to organize, as in tarot The tarot , pronounced /ˈtæroʊ/, is a pack of cards (most commonly numbering 78), used from the mid-15th century in various parts of Europe to play card games such as Italian tarocchini and French tarot. From the late 18th century until the present time the tarot has also found use by mystics and occultists in efforts at divination or as a map, the I Ching The I Ching , "Yì Jīng" (Pinyin), also known as the Book of Changes, Classic of Changes; and Zhouyi, is one of the oldest of the Chinese classic texts. The book contains a divination system comparable to Western geomancy or the West African Ifá system. In Western cultures and modern East Asia, it is still widely used for this purpose, and 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:. 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 is the attempt to gain insight into a question or situation by way of a standardized process or ritual. Diviners ascertain their interpretations of how a querent should proceed by reading signs, events, or omens, or through alleged contact with a supernatural agency. Divination can be seen as a systematic method with which to organize to attempt to circumvent randomness and fate.^[3][4]

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 is a branch in 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 of calculus. Calculus is the study of change, in had a positive impact on the formal study of randomness. In the 1888 edition of his book The Logic of Chance John Venn John Venn FRS , 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 π is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.141593 in the usual decimal notation. Many formulae from mathematics, science, and engineering involve π, which by using them to construct a random walk in two dimensions.^[5]

The early part of the twentieth century saw a rapid growth in the formal analysis of randomness, as various approaches for a mathematical foundations of probability were introduced. In the mid to late twentieth century ideas of 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. According to Gregory Chaitin, it is "the result of putting Shannon's information theory and Turing's computability theory into a cocktail shaker and shaking vigorously." introduced new dimensions to the field via the concept of algorithmic randomness In algorithmic information theory , the Kolmogorov complexity (also known as descriptive complexity, Kolmogorov-Chaitin complexity, stochastic complexity, algorithmic entropy, or program-size complexity) of an object such as a piece of text is a measure of the computational resources needed to specify the object. For example, consider the.

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 A randomized algorithm or probabilistic algorithm is an algorithm which employs a degree of randomness as part of its logic. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random bits. Formally, outperform the best deterministic methods.

Randomness in science

Many scientific fields are concerned with randomness:

• Algorithmic probability Algorithmic probability is a concept in theoretical computer science; it quantifies the idea of theories and predictions with reference to short programs and their output. Around 1960, Ray Solomonoff invented the concept of algorithmic probability: take a universal computer and randomly generate an input program. The program will compute some
• Chaos theory Chaos theory is a field of study in mathematics, physics, economics and philosophy studying the behavior of dynamical systems that are highly sensitive to initial conditions. This sensitivity is popularly referred to as the butterfly effect. Small differences in initial conditions yield widely diverging outcomes for chaotic systems, rendering long-
• Cryptography Cryptography is the practice and study of hiding information. Modern cryptography intersects the disciplines of mathematics, computer science, and engineering. Applications of cryptography include ATM cards, computer passwords, and electronic commerce
• Game theory Game theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology , engineering, political science, international relations, computer science, and philosophy. Game theory attempts to mathematically capture behavior in strategic situations, or games, in which an individual's success
• Information theory Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Historically, 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 communicating data. Since its inception it
• Pattern recognition Pattern recognition is "the act of taking in raw data and taking an action based on the category of the pattern". Most research in pattern recognition is about methods for supervised learning and unsupervised learning
• 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 occurrences or evolve over time in an apparently random
• 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 scales. In advanced topics of QM, some of these
• Statistics Statistics is the formal science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments
• Statistical mechanics Statistical mechanics is the application of probability theory (which contains mathematical tools for dealing with large populations) to study the thermodynamic behavior of systems of a large number of particles. It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties

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 is the application of probability theory (which contains mathematical tools for dealing with large populations) to study the thermodynamic behavior of systems of a large number of particles. It provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk properties in order to explain phenomena in thermodynamics In science, thermodynamics is the study of energy conversion between heat and mechanical work, and subsequently the macroscopic variables such as temperature, volume and pressure and the properties of gases This article outlines the historical development of the laws describing ideal gases. For a detailed description of the ideal gas laws and their further development, see Ideal gas, Ideal gas law and Gas.

According to several standard interpretations of 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 scales. In advanced topics of QM, some of these, microscopic phenomena are objectively random^{[citation needed]}. 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 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 . The electrons of an atom are bound to the nucleus by the electromagnetic force. Likewise, a group of atoms can remain 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.^[6] Thus, quantum mechanics does not specify the outcome of individual experiments but only the probabilities. Hidden variable theories Historically, in physics, hidden variable theories were espoused by a minority of physicists who argued that the statistical nature of quantum mechanics indicated that quantum mechanics is "incomplete". Albert Einstein, the most famous proponent of hidden variables, insisted that, "I am convinced God does not play dice" — 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 The modern evolutionary synthesis is a union of ideas from several biological specialties which forms a logical account of evolution. This synthesis has been accepted by nearly all working biologists. The synthesis was produced over about a decade (1936–1947), and the development of population genetics (1918–1932) was the stimulus. This showed ascribes the observed diversity of life to natural selection Natural selection is a natural law by which genetically heritable traits become more or less common in a population over successive generations. This selection in interaction with the production of variation, the possible genetic fixation process and possibly, in several cases, whith little epigenetic process determine the evolution of the species, in which some random genetic mutations Mutations are changes in a genomic sequence: the DNA sequence of a cell's genome or the DNA or RNA sequence of a virus. Mutations are caused by radiation, viruses, transposons and mutagenic chemicals, as well as errors that occur during meiosis or DNA replication. They can also be induced by the organism itself, by cellular processes such as are retained in the gene pool In population genetics, a gene pool is the complete set of unique alleles in a species or population 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 Freckles are clusters of concentrated melanin which are most often visible on people with a fair complexion. A freckle is also called an "ephelis" 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.^[7]

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 a way of expressing knowledge or belief that an event will occur or has occurred. The concept has been given an exact mathematical meaning in probability theory, which is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, and philosophy to draw conclusions about the likelihood of arose from attempts to formulate mathematical descriptions of chance events, originally in the context of 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. Typically, the outcome of the wager is evident within a short period, but later in connection with physics. Statistics Statistics is the formal science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments is used to infer the underlying probability distribution of a collection of empirical observations. For the purposes of simulation, it is necessary to have a large supply of random numbers or means to generate them on demand.

Algorithmic information theory studies, among other topics, what constitutes a random sequence. The central idea is that a string of bits 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. Pioneers of this field include Andrey Kolmogorov and his student Per Martin-Löf, Ray Solomonoff, and Gregory Chaitin.

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 only eight possible values:

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

Therefore, as the random sequence progresses, it must recycle through 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) and Pi. 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, 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.^[8]

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, 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 finance

The random walk hypothesis considers that asset prices in an organized market 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.

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