Randomness

From Wikipedia, the free encyclopedia

“Random” redirects here. For other uses, see Random (disambiguation).

The word random is used to express lack of purpose, cause, order, 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 randomness is often used in statistics to signify well defined statistical properties, such as lack of bias or correlation.

Randomness has an important place in science, philosophy, and religion.

[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 western 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:

[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. ^[1]Thus quantum mechanics does not specify the outcome of individual experiments but only the probabilities. Hidden variable theories attempt to escape the view that nature contains irreducible randomness: such theories posit that in the processes that appear random, unobservable (hidden) 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, genes and exposure to light only control the density of freckles that appear on a person's skin; whereas the exact location of individual freckles appears to be random^{[citation needed]}.

Note that this effect isn't limited to physical characteristics. Sexual orientation also appears to have a random element, for example. In identical twin studies, such twins are more likely to have the same sexual orientation than two randomly chosen individuals in any given population. This correlation is attributable to genetics and chemical influences within the womb if the twins are adopted and raised in separate environments, but could be due to either genetic or environmental factors if they are raised in the same environment. However, even identical twins raised in the same environment do not always have the same sexual orientation. In cases where there is a difference in sexual orientation between the two, this is typically ascribed to a random element, although this could also result from a pattern of events more complex than is currently understood^{[citation needed]}.

[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, Ray Solomonoff, Gregory Chaitin, Anders Martin-Löf, 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 which who has the cryptographic key needed to turn the sequence of bits into a readable message. The message is not random, but is for one of the observers unpredictable. This is a key point indicating that there is order behind all that appears to be random.

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.

Some mathematically defined sequences exhibit some of the same characteristics as random sequences, but because they are generated by a describable mechanism they are called pseudo-random.

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.

It is important to remember that the randomness of a phenomenon is not itself random and 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. Human, acting based on free will, have thoughts that often lead to actions that occur in the physical universe. Therefore, free will is potentially a means that interjects random action into the natural universe.

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 what the King James Version of the Bible is telling humans. 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.

Further inspection into the origins of Judeo/Christian religion indicates one view that there is a very strict understanding of predestination excluding any possibility of random events. At the time of Christ the Qumran, a tribe outside of Jerusalem by the Dead Sea, had scrolls that documented their very strict deterministic worldview. Elements of this worldview are found in modern Christianity. For example, the King James and NIV Bible tells humans that God knew the believers before the foundations of time, ECC 3 is about God's perfect timing, and Daniel, Ezkiel, and Revelation tell humans that the end state is already determined. Moreover, the Judeo\Christian Bible indicates that there is a purpose to everything which is found in ECC 3 also.

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 a foundation for Intelligent Design which is counter to Evolution that remarks the natural emerges based on random selection.

Donald Knuth, a stanford computer scientist and Christian commentator, remarks that he finds psuedo-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 peoples 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.^[2]

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]}

Matt Ridley, a zoology doctorate and science writer, writes how humans, a paradoxical creature, can be simultaneously free-willed and motivated by instinct and culture. Ridley suggests that experience and genes have interplay. In his writings he explores DNA as a pattern makers template, not as a blueprint for life, and points to causes of free will as consequences to genetic outcomes. Ridley, in his musings, suggests that the evolutionary force he thinks shapes the contents of our genes is the Genome Organizing Device, or GOD. In Ridley's mind GOD is the pattern maker. Thus, Ridley literally defies natural selection indicating that sources of randomness include God and human decision making.^{[citation needed]}

[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 random events 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 :

Randomness coming from the environment (for example, Brownian motion, but also hardware random number generators)
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 ...).
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 amount 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.^[3]

[edit] Links related to generating randomness

[edit] Misconceptions/logical fallacies

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. For a more detailed discussion, see Gambler's fallacy.

[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's 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"

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 only valid if the roll is somehow biased and results don't have equal probabilities - for example, with weighted dice. If we know for certain that the roll is fair, then previous events have no influence over 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] References

^ "Each nucleus decays spontaneously, at random, in accordance with the blind workings of chance". Q for Quantum, John Gribbin
^ Donald Knuth, "Things A Computer Scientist Rarely Talks About", Pg 185, 190-191, CSLI
^ Terry Ritter, Randomness tests: a literature survey. http://www.ciphersbyritter.com/RES/RANDTEST.HTM

[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

[edit] External links

Look up randomness in Wiktionary, the free dictionary.

Wikiquote has a collection of quotations related to:

Randomness

Random.org generates random numbers
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
The Random Forum Forum of randomness
RandomText.net Generates Random Text

Retrieved from "http://en.wikipedia.org../../../r/a/n/Randomness.html"