Randomness

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Randomness is a lack of order, purpose, cause, or predictability in non-scientific parlance. A random process is a repeating process whose outcomes follow no describable deterministic pattern, but follow a probability distribution.

The term is often used in statistics to signify well defined statistical properties, such as lack of bias or correlation. Monte Carlo Methods, which rely on random input, are important techniques of computational science.^[1] Random selection is an official method to resolve tied elections in some jurisdictions^[2], and is even an ancient method of divination, as in tarot, the I Ching, and bibliomancy.

Contents 1 History 2 Randomness in science 2.1 In the physical sciences 2.2 In biology 2.3 In mathematics 2.4 In information science 2.5 In finance 2.6 Randomness versus unpredictability 3 Randomness and religion 4 Applications and use of randomness 4.1 Generating randomness 4.2 Randomness measures and tests 4.3 Links related to generating randomness 5 Misconceptions/logical fallacies 5.1 A number is "due" 5.2 A number is "cursed" 6 Books 7 See also 8 References 9 External links

[edit] History

Humankind has been concerned with random physical processes since pre-historic times. Examples are divination (cleromancy, reading messages in casting lots), the use of allotment in the Athenian democracy, and the frequent references to the casting of lots found in the Old Testament.

Despite the prevalence of gambling in all times and cultures, for a long time there was little inquiry into the subject. Though Gerolamo Cardano and Galileo wrote about games of chance, the first mathematical treatments were given by Blaise Pascal, Pierre de Fermat and Christiaan Huygens. The classical version of probability theory that they developed proceeds from the assumption that outcomes of random processes are equally likely; thus they were among the first to give a definition of randomness in statistical terms. The concept of statistical randomness was later developed into the concept of information entropy in information theory.

In the early 1960s Gregory Chaitin, Andrey Kolmogorov and Ray Solomonoff introduced the notion of algorithmic randomness, in which the randomness of a sequence depends on whether it is possible to compress it.

[edit] Randomness in science

Many scientific fields are concerned with randomness:

• Algorithmic probability
• Chaos theory
• Cryptography
• Game theory
• Information theory
• Pattern recognition
• Probability theory
• Quantum mechanics
• Statistics
• Statistical mechanics

[edit] In the physical sciences

In the 19th century scientists used the idea of random motions of molecules in the development of statistical mechanics in order to explain phenomena in thermodynamics and the properties of gases.

According to several standard interpretations of quantum mechanics, 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 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.^[3] Thus quantum mechanics does not specify the outcome of individual experiments but only the probabilities. Hidden variable theories 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.

[edit] In biology

The theory of evolution ascribes the observed diversity of life to random genetic mutations some of which are retained in the gene pool due to the 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.^[4]

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.

[edit] In mathematics

The mathematical theory of probability arose from attempts to formulate mathematical descriptions of chance events, originally in the context of gambling but soon in connection with situations of interest in physics. Statistics 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 basically 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, Gregory Chaitin, and others.

[edit] 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.

[edit] 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.

[edit] Randomness versus unpredictability

Randomness is an objective property. Nevertheless, what appears random to one observer may not appear random to another observer. Consider two observers of a sequence of bits, only one of whom has the cryptographic key needed to turn the sequence of bits into a readable message. The message is not random, but is unpredictable for one of the observers. One of the intriguing aspects of random processes is that it is hard to know whether the process is truly random. The observer can always suspect that there is some "key" that unlocks the message. This is one of the foundations of superstition and is also what is a driving motive, curiosity, for discovery in science and mathematics.

Under the cosmological hypothesis of determinism there is no randomness in the universe, only unpredictability, since there is only one possible outcome to all events in the universe. No event under determinism can be defined as having probability since again there is only one universal outcome.

Some mathematically defined sequences, such as the decimals of pi, 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 are unpredictable in practice due to their extreme dependence on 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 turns out to be identified with computational unpredictability.

Randomness of a phenomenon is not itself 'random'. It can often be precisely characterized, 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 a random one. More simply, though we cannot predict the outcome of a single toss of a fair coin, we can characterize its general behavior by saying that if a large number of tosses are made, roughly half of them will show up "Heads". Ohm's law and the kinetic theory of gases are precise characterizations of macroscopic phenomena which are random on the microscopic level.

[edit] Randomness and religion

Randomness has been associated closely with the notion of free will in a number of ways. If a person has free will (as defined by incompatibilists), then his actions will be unpredictable by other people and will contain an element of irreducible indeterminacy. By religious or supernatural conceptions of incompatibilist free will, such human actions may be the only source of randomness in the universe. (According to the naturalistic conception, by contrast, incompatibilist free will arises from pre-existing indeterminacy in physical laws and is not necessarily a unique feature of humans. According to the compatibilist conception, there is no randomness and humans are merely too complex to be easily predicted).

Some theologians have attempted to resolve the apparent contradiction between an omniscient deity, or a first cause, and free will using randomness. Discordians have a strong belief in randomness and unpredictability. Buddhist philosophy states that any event is the result of previous events (karma) and as such there is no such thing as a random event nor a 'first' event.

Martin Luther, the forefather of Protestantism, believed that there was nothing random based on his understanding of the Bible. 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, where, according to theory, (non-random) selection is applied to the results of random genetic variation.

Donald Knuth, a Stanford computer scientist and Christian commentator, remarks that he finds pseudo-random 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.^[5]

C. S. Lewis, 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…" Lewis, believing in free will, had an indirect belief in randomness by setting up a dependency of love on free will.^{[citation needed]}

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.

[edit] Applications and use of randomness

Main article: Applications of randomness

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

Political: Greek Democracy was based on the concept of isonomia (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, such as selecting jurors and military draft lotteries.

Social: Random numbers were first investigated in the context of gambling, and many randomizing devices such as dice, shuffling playing cards, and roulette wheels, were first developed for use in gambling. The ability to fairly produce random numbers is vital to electronic gambling and, as such, the methods used to create them are usually regulated by government Gaming Control Boards. 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).

Mathematical: Random numbers are also used where their use is mathematically important, such as sampling for opinion polls and for statistical sampling in quality control systems. Computational solutions for some types of problems use random numbers extensively, such as in the Monte Carlo method and in genetic algorithms.

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 such as cleromancy see what appears to be a random event as a means for a divine being to communicate their will. (See also Free will and Determinism).

[edit] Generating randomness

Main article: Random number generation

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, but also hardware random number generators)
2. Randomness coming from the initial conditions. This aspect is studied by chaos theory, and is observed in systems whose behavior is very sensitive to small variations in initial conditions (such as pachinko machines, dice ...).
3. Randomness intrinsically generated by the system. This is also called pseudorandomness, and is the kind used in pseudo-random number generators. There are many algorithms (based on arithmetics or cellular automaton) to generate pseudorandom numbers. The behavior of the system can be determined by knowing the seed state and the algorithm used. These methods are quicker than getting "true" randomness from the environment.

The many applications of randomness 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 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 tables.

[edit] Randomness measures and tests

There are many practical measures of randomness for a binary sequence. These include measures based on frequency, discrete transforms, and complexity or a mixture of these. These include tests by Kak, Phillips, Yuen, Hopkins, Beth and Dai, Mund, and Marsaglia and Zaman.^[6]

[edit] Links related to generating randomness

• Hardware random number generator
• Entropy (computing)
• Information entropy
• Probability theory
• Pseudorandomness
• Pseudorandom number generator
• Random number
• Random sequence
• Random variable
• Randomization
• Stochastic process
• White noise

[edit] Misconceptions/logical fallacies

Main article: Gambler's fallacy

Popular perceptions of randomness are frequently wrong, based on logical fallacies. The following is an attempt to identify the source of such fallacies and correct the logical errors.

[edit] A number is "due"

This argument says that "since all numbers will eventually appear in a random selection, 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 cards are drawn and not returned to the deck. It is true, for example, that 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, there is an equal chance of drawing a jack or any other card the next time. The same truth applies to any other case where objects are selected independently and nothing is removed from the system after each event, such as a die roll, coin toss or most lottery number selection schemes. A way to look at it is to note that random processes such as throwing coins don't have memory, making it impossible for past outcomes to affect the present and future.

[edit] A number is "cursed"

See also: Benford's law

This argument is almost the reverse of the above, and says that numbers which have come up less often in the past will continue to come up less often in the future. A similar "number is 'blessed'" argument might be made saying that numbers which have come up more often in the past are likely to do so in the future. This logic is valid if and only if the roll might be somehow biased — for example, with weighted dice. If we know for certain that the roll is fair, then previous events give no indication of future events.

Note that in nature, unexpected or uncertain events rarely occur with perfectly equal frequencies, so learning which events are likely to have higher probability by observing outcomes makes sense. What is fallacious is to apply this logic to systems which are specially designed so that all outcomes are equally likely — such as dice, roulette wheels, and so on.

[edit] 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, Reading, MA: Addison-Wesley, 1997. ISBN 0-201-89684-2
• Fooled by Randomness, 2nd ed. by Nassim Nicholas Taleb. Thomson Texere, 2004. ISBN 1-58799-190-X
• Exploring Randomness by Gregory Chaitin. Springer-Verlag London, 2001. ISBN 1-85233-417-7
• Random, by Kenneth Chan, includes a "Random Scale" for grading the level of randomness

[edit] See also

• Aleatory
• Allotment
• Complexity
• Chaos
• Probability interpretations
• Random number generator
• Randomness tests
• Frequency probability
• Chaitin's constant
• Algorithmic information theory
• Algorithmic probability

[edit] References

[edit] External links

• Random.org generates random numbers
• HotBits generates random numbers from radioactive decay
• Yuzoz.com generates random numbers from live space events
• Chaitin: Randomness and Mathematical Proof
• A Pseudorandom Number Sequence Test Program (Public Domain)
• Dictionary of the History of Ideas: Chance
• Philosophy: Free Will vs. Determinism
• RAHM Nation Institute
• History of randomness definitions, in Stephen Wolfram's A New Kind of Science.
• Computing a Glimpse of Randomness