12/12/2014 Randomness ­ Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Randomness 1/11

RandomnessFrom Wikipedia, the free encyclopedia

Randomness means lack of pattern or predictability in events.[1] Randomness suggests a non­order ornon­coherence in a sequence of symbols or steps, such that there is no intelligible pattern orcombination.

Random events are individually unpredictable, but the frequency of different outcomes over a largenumber of events (or "trials") are frequently predictable. For example, when throwing two dice andcounting the total, a sum of 7 will randomly occur twice as often as 4, but the outcome of any particularroll of the dice is unpredictable. This view, where randomness simply refers to situations where thecertainty of the outcome is at issue, applies to concepts of chance, probability, and information entropy.In these situations, randomness implies a measure of uncertainty, and notions of haphazardness areirrelevant.

The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, arandom variable is an assignment of a numerical value to each possible outcome of an event space. Thisassociation facilitates the identification and the calculation of probabilities of the events. A randomprocess is a sequence of random variables describing a process whose outcomes do not follow adeterministic pattern, but follow an evolution described by probability distributions. These and otherconstructs are extremely useful in probability theory.

Randomness is often used in statistics to signify well­defined statistical properties. Monte Carlomethods, which rely on random input, are important techniques in science, as, for instance, incomputational science.[2]

Random selection is a method of selecting items (often called units) from a population where theprobability of choosing a specific item is the proportion of those items in the population. For example, ifwe have a bowl of 100 marbles with 10 red (and any red marble is indistinguishable from any other redmarble) and 90 blue (and any blue marble is indistinguishable from any other blue marble), a randomselection mechanism would choose a red marble with probability 1/10. Note that a random selectionmechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. Insituations where a population consists of items that are distinguishable, a random selection mechanismrequires equal probabilities for any item to be chosen. That is, if the selection process is such that eachmember of a population, of say research subjects, has the same probability of being chosen then we cansay the selection process is random.

Contents

1 History2 Randomness in science

2.1 In the physical sciences2.2 In biology2.3 In mathematics2.4 In statistics2.5 In information science

12/12/2014 Randomness ­ Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Randomness 2/11

Ancient fresco of dice players inPompei.

2.6 In finance2.7 Randomness versus unpredictability

3 Randomness and politics4 Randomness and religion5 Applications and use of randomness

5.1 Generating randomness5.2 Randomness measures and tests

6 Misconceptions and logical fallacies6.1 A number is "due"6.2 A number is "cursed" or "blessed"6.3 Odds are never dynamic

7 See also8 References9 Further reading10 External links

History

In ancient history, the concepts of chance and randomness wereintertwined with that of fate. Many ancient peoples threw dice todetermine fate, and this later evolved into games of chance. Mostancient cultures used various methods of divination to attempt tocircumvent randomness and fate.[3][4]

The Chinese were perhaps the earliest people to formalize oddsand chance 3,000 years ago. The Greek philosophers discussedrandomness at length, but only in non­quantitative forms. It wasonly in the sixteenth century that Italian mathematicians began toformalize the odds associated with various games of chance. Theinvention of the calculus had a positive impact on the formalstudy of randomness. In the 1888 edition of his book The Logicof Chance John Venn wrote a chapter on The conception ofrandomness that included his view of the randomness of the digits of the number Pi by using them toconstruct a random walk in two dimensions.[5]

The early part of the twentieth century saw a rapid growth in the formal analysis of randomness, asvarious approaches to the mathematical foundations of probability were introduced. In the mid­ to late­twentieth century, ideas of algorithmic information theory introduced new dimensions to the field via theconcept of algorithmic randomness.

Although randomness had often been viewed as an obstacle and a nuisance for many centuries, in thetwentieth century computer scientists began to realize that the deliberate introduction of randomnessinto computations can be an effective tool for designing better algorithms. In some cases suchrandomized algorithms outperform the best deterministic methods.

12/12/2014 Randomness ­ Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Randomness 3/11

Randomness in science

Many scientific fields are concerned with randomness:

In the physical sciences

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

According to several standard interpretations of quantum mechanics, microscopic phenomena areobjectively random.[6] That is, in an experiment that controls all causally relevant parameters, someaspects of the outcome still vary randomly. For example, if you place a single unstable atom in acontrolled environment, you cannot predict how long it will take for the atom to decay—only theprobability of decay in a given time.[7] Thus, quantum mechanics does not specify the outcome ofindividual experiments but only the probabilities. Hidden variable theories reject the view that naturecontains irreducible randomness: such theories posit that in the processes that appear random, propertieswith a certain statistical distribution are at work behind the scenes, determining the outcome in eachcase.

In biology

The modern evolutionary synthesis ascribes the observed diversity of life to natural selection, in whichsome random genetic mutations are retained in the gene pool due to the systematically improved chancefor 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 ofgenes and the environment) and to some extent randomly. For example, the density of freckles thatappear on a person's skin is controlled by genes and exposure to light; whereas the exact location ofindividual freckles seems random.[8]

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 pursuingpredators to predict their trajectories.

In mathematics

Algorithmic probabilityChaos theoryCryptographyGame theoryInformation theoryPattern recognitionProbability theoryQuantum mechanicsStatistical mechanicsStatistics

12/12/2014 Randomness ­ Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Randomness 4/11

The mathematical theory of probability arose from attempts to formulate mathematical descriptions ofchance events, originally in the context of gambling, but later in connection with physics. Statistics isused to infer the underlying probability distribution of a collection of empirical observations. For thepurposes of simulation, it is necessary to have a large supply of random numbers or means to generatethem on demand.

Algorithmic information theory studies, among other topics, what constitutes a random sequence. Thecentral idea is that a string of bits is random if and only if it is shorter than any computer program thatcan produce that string (Kolmogorov randomness)—this means that random strings are those that cannotbe compressed. Pioneers of this field include Andrey Kolmogorov and his student Per Martin­Löf, RaySolomonoff, and Gregory Chaitin.

In mathematics, there must be an infinite expansion of information for randomness to exist. This canbest be seen with an example. Given a random sequence of three­bit numbers, each number can haveone of only eight possible values:

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

Therefore, as the random sequence progresses, it must recycle previous values. To increase theinformation space, another bit may be added to each possible number, giving 16 possible values fromwhich to pick a random number. It could be said that the random four­bit number sequence is morerandom than the three­bit one. This suggests that true randomness requires an infinite expansion of theinformation space.

Randomness occurs in numbers such as log (2) and pi. The decimal digits of pi constitute an infinitesequence and "never repeat in a cyclical fashion." Numbers like pi are also considered likely to benormal, which means 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 ofthe 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 inother number bases.[9]

In statistics

In statistics, randomness is commonly used to create simple random samples. This lets surveys ofcompletely random groups of people provide realistic data. Common methods of doing this includedrawing names out of a hat or using a random digit chart. A random digit chart is simply a large table ofrandom digits.

In information science

In information science, irrelevant or meaningless data is considered noise. Noise consists of a largenumber 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 ofits variation that is causally attributable to the source, the signal.

12/12/2014 Randomness ­ Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Randomness 5/11

In terms of the development of random networks, for communication randomness rests on the twosimple assumptions of Paul Erdős and Alfréd Rényi who said that there were a fixed number of nodesand this number remained fixed for the life of the network, and that all nodes were equal and linkedrandomly to each other.[10]

In finance

The random walk hypothesis considers that asset prices in an organized market evolve at random, in thesense that the expected value of their change is zero but the actual value may turn out to be positive ornegative. More generally, asset prices are influenced by a variety of unpredictable events in the generaleconomic environment.

Randomness versus unpredictability

Randomness, as opposed to unpredictability, is an objective property. Determinists believe it is anobjective fact that randomness does not in fact exist. Also, what appears random to one observer maynot appear random to another. Consider two observers of a sequence of bits, when only one of whom hasthe cryptographic key needed to turn the sequence of bits into a readable message. For that observer themessage 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 trulyrandom. An observer may suspect that there is some "key" that unlocks the message. This is one of thefoundations of superstition, but also a motivation for discovery in science and mathematics.

Under the cosmological hypothesis of determinism, there is no randomness in the universe, onlyunpredictability, since there is only one possible outcome to all events in the universe. A follower of thenarrow frequency interpretation of probability could assert that no event can be said to have probability,since there is only one universal outcome. Under the rival Bayesian interpretation of probability, there isno objection to using probabilities to represent a lack of complete knowledge of outcomes.

Some mathematically defined sequences, such as the decimals of pi mentioned above, exhibit some ofthe same characteristics as random sequences, but because they are generated by a describablemechanism, they are called pseudorandom. To an observer who does not know the mechanism, apseudorandom sequence is unpredictable.

Chaotic systems 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 withcomputational unpredictability.

Individual events that are random may still be precisely described en masse, usually in terms ofprobability or expected value. For instance, quantum mechanics allows a very precise calculation of thehalf­lives of atoms even though the process of atomic decay is random. More simply, although a singletoss of a fair coin cannot be predicted, its general behavior can be described by saying that if a largenumber of tosses are made, roughly half of them will show up heads. Ohm's law and the kinetic theoryof gases are non­random macroscopic phenomena that are assumed random at the microscopic level.

Randomness and politics

Random selection can be an official method to resolve tied elections in some jurisdictions.[11] Its use inpolitics is very old, as office holders in Ancient Athens were chosen by lot, there being no voting.

12/12/2014 Randomness ­ Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Randomness 6/11

Randomness and religion

Randomness can be seen as conflicting with the deterministic ideas of some religions, such as thosewhere the universe is created by an omniscient deity who is aware of all past and future events. If theuniverse is regarded to have a purpose, then randomness can be seen as impossible. This is one of therationales for religious opposition to evolution, which states that non­random selection is applied to theresults of random genetic variation.

Hindu and Buddhist philosophies state that any event is the result of previous events, as reflected in theconcept of karma, and as such there is no such thing as a random event or a first event.

In some religious contexts, procedures that are commonly perceived as randomizers are used fordivination. Cleromancy uses the casting of bones or dice to reveal what is seen as the will of the gods.

Followers of Discordianism, who venerate Eris the Greco­Roman goddess of chaos, have a strong beliefin randomness and unpredictability.

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 was based on the concept of isonomia (equality of political rights) andused complex allotment machines to ensure that the positions on the ruling committees that ran Athenswere fairly allocated. Allotment is now restricted to selecting jurors in Anglo­Saxon legal systems and insituations where "fairness" is approximated by randomization, such as selecting jurors and military draftlotteries.

Social: Random numbers were first investigated in the context of gambling, and many randomizingdevices, such as dice, shuffling playing cards, and roulette wheels, were first developed for use ingambling. The ability to produce random numbers fairly is vital to electronic gambling, and, as such, themethods used to create them are usually regulated by government Gaming Control Boards. Randomdrawings are also used to determine lottery winners. Throughout history, randomness has been used forgames of chance and to select out individuals for an unwanted task in a fair way (see drawing straws).

Sports: Some sports, including American Football, use coin tosses to randomly select starting conditionsfor games or seed tied teams for postseason play. The National Basketball Association uses a weightedlottery to order teams in its draft.

Mathematical: Random numbers are also used where their use is mathematically important, such assampling for opinion polls and for statistical sampling in quality control systems. Computationalsolutions for some types of problems use random numbers extensively, such as in the Monte Carlomethod 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 seewhat appears to be a random event as a means for a divine being to communicate their will. (See alsoFree will and Determinism).

12/12/2014 Randomness ­ Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Randomness 7/11

The ball in a roulette can be usedas a source of apparentrandomness, because its behavioris very sensitive to the initialconditions.

Generating randomness

It is generally accepted that there exist three mechanismsresponsible for (apparently) random behavior in systems:

1. Randomness coming from the environment (for example,Brownian motion, but also hardware random numbergenerators)

2. Randomness coming from the initial conditions. This aspect isstudied by chaos theory and is observed in systems whosebehavior is very sensitive to small variations in initialconditions (such as pachinko machines and dice).

3. Randomness intrinsically generated by the system. This is alsocalled pseudorandomness and is the kind used in pseudo­random number generators. There are many algorithms (based on arithmetics or cellularautomaton) to generate pseudorandom numbers. The behavior of the system can be determined byknowing the seed state and the algorithm used. These methods are often 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 theycan generate random numbers.

Before the advent of computational random number generators, generating large amounts of sufficientlyrandom numbers (important in statistics) required a lot of work. Results would sometimes be collectedand distributed as random number tables.

Randomness measures and tests

There are many practical measures of randomness for a binary sequence. These include measures basedon 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.[12]

Misconceptions and logical fallacies

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

A number is "due"

This argument is, "In a random selection of numbers, since all numbers eventually appear, those thathave not come up yet are 'due', and thus more likely to come up soon." This logic is only correct ifapplied to a system where numbers that come up are removed from the system, such as when playingcards are drawn and not returned to the deck. In this case, once a jack is removed from the deck, the nextdraw is less likely to be a jack and more likely to be some other card. However, if the jack is returned tothe deck, and the deck is thoroughly reshuffled, a jack is as likely to be drawn as any other card. Thesame applies in any other process where objects are selected independently, and none are removed after

12/12/2014 Randomness ­ Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Randomness 8/11

When the hostreveals that onedoor onlycontained a goat,this is newinformation.

each event, such as the roll of a die, a coin toss, or most lottery number selection schemes. Truly randomprocesses such as these do not have memory, making it impossible for past outcomes to affect futureoutcomes.

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 oftenin the past, and so it is thought that it will occur less often in the future. A number may be assumed to beblessed because it has occurred more often than others in the past, and so it is thought likely to come upmore often in the future. This logic is valid only if the randomisation is biased, for example with aloaded 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 whichevents are more probable makes sense. It is fallacious to apply this logic to systems designed to make alloutcomes 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 onegains more information about that situation, they may need to re­calculate the odds.

If we are told that a woman has two children, and one of them is a girl, what are theodds 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 usingmathematician Gerolamo Cardano's method of building a Probability space(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 havingthese two children: boy­boy, girl­boy, boy­girl, and girl­girl. But we were givenmore information. Once we are told that one of the children is a female, we use thisnew information to eliminate the boy­boy scenario. Thus the probability spacereveals that there are still 3 ways to have two children where one is a female: boy­girl, girl­boy, girl­girl. Only 1/3 of these scenarios would have the other child alsobe a girl.[13] Using a probability space, we are less likely to miss one of the possible scenarios, or toneglect the importance of new information. For further information, see Boy or girl paradox.

This technique provides insights in other situations such as the Monty Hall problem, a game showscenario in which a car is hidden behind one of three doors, and two goats are hidden as booby prizesbehind the others. Once the contestant has chosen a door, the host opens one of the remaining doors toreveal a goat, eliminating that door as an option. With only two doors left (one with the car, the otherwith 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 choosingbetween 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 theirchances of winning by changing to the other door.[13]

See also

Algorithmic probabilityAleatoryChaitin's constantChance (disambiguation)

12/12/2014 Randomness ­ Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Randomness 9/11

References

Chance (disambiguation)Chaos theoryCryptographyFrequency probabilityGame theoryInformation theoryNonlinear systemPattern recognitionPredictabilityProbability interpretationsProbability theoryPseudorandomnessQuantum mechanicsStatistical mechanicsStatistics

1. ^ The Oxford English Dictionary defines "random" as "Having no definite aim or purpose; not sent or guidedin a particular direction; made, done, occurring, etc., without method or conscious choice; haphazard."

2. ^ Third Workshop on Monte Carlo Methods(http://www.people.fas.harvard.edu/~junliu/Workshops/workshop2007/), Jun Liu, Professor of Statistics,Harvard University

3. ^ Handbook to life in ancient Rome by Lesley Adkins 1998 ISBN 0­19­512332­8 page 2794. ^ Religions of the ancient world by Sarah Iles Johnston 2004 ISBN 0­674­01517­7 page 3705. ^ Annotated readings in the history of statistics by Herbert Aron David, 2001 ISBN 0­387­98844­0 page 115.

Note that the 1866 edition of Venn's book (on Google books) does not include this chapter.6. ^ Nature.com (http://www.nature.com/nature/journal/v446/n7138/abs/nature05677.html) in Bell's aspect

experiment: Nature7. ^ "Each nucleus decays spontaneously, at random, in accordance with the blind workings of chance." Q forQuantum, John Gribbin

8. ^ Breathnach, A. S. (1982). "A long­term hypopigmentary effect of thorium­X on freckled skin". BritishJournal of Dermatology 106 (1): 19–25. doi:10.1111/j.1365­2133.1982.tb00897.x(http://dx.doi.org/10.1111%2Fj.1365­2133.1982.tb00897.x). PMID 7059501(https://www.ncbi.nlm.nih.gov/pubmed/7059501). "The distribution of freckles seems entirely random, andnot associated with any other obviously punctuate anatomical or physiological feature of skin."

9. ^ "Are the digits of pi random? researcher may hold the key" (http://www.lbl.gov/Science­Articles/Archive/pi­random.html). Lbl.gov. 2001­07­23. Retrieved 2012­07­27.

10. ^ Laszso Barabasi, (2003), Linked, Rich Gets Richer, P8111. ^ Municipal Elections Act (Ontario, Canada) 1996, c. 32, Sched., s. 62 (3) : "If the recount indicates that two

or more candidates who cannot both or all be declared elected to an office have received the same number ofvotes, the clerk shall choose the successful candidate or candidates by lot."

12. ^ Terry Ritter, Randomness tests: a literature survey. ciphersbyritter.com(http://www.ciphersbyritter.com/RES/RANDTEST.HTM)

^ a b Johnson, George (8 June 2008). "Playing the Odds"

12/12/2014 Randomness ­ Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Randomness 10/11

Wikiversity has learningmaterials about Random

Look up randomness inWiktionary, the freedictionary.

Wikiquote has quotationsrelated to: Randomness

Wikimedia Commons hasmedia related toRandomness.

Further reading

Randomness by Deborah J. Bennett. Harvard University Press, 1998. ISBN 0­674­10745­4.Random Measures, 4th ed. by Olav Kallenberg. Academic Press, New York, London; Akademie­Verlag, Berlin, 1986. MR0854102.The Art of Computer Programming. Vol. 2: Seminumerical Algorithms, 3rd ed. by Donald E.Knuth. Reading, MA: Addison­Wesley, 1997. ISBN 0­201­89684­2.Fooled by Randomness, 2nd ed. by Nassim Nicholas Taleb. Thomson Texere, 2004. ISBN 1­58799­190­X.Exploring Randomness by Gregory Chaitin. Springer­Verlag London, 2001. ISBN 1­85233­417­7.Random by Kenneth Chan includes a "Random Scale" for grading the level of randomness.The Drunkard’s Walk: How Randomness Rules our Lives by Leonard Mlodinow. Pantheon Books,New York, 2008. ISBN 978­0­375­42404­5.

External links

An 8­foot­tall (2.4 m) Probability Machine (named SirFrancis) comparing stock market returns to the randomnessof the beans dropping through the quincunx pattern.(https://www.youtube.com/watch?v=AUSKTk9ENzg) onYouTube from Index Funds Advisors IFA.com(http://www.ifa.com)QuantumLab (http://www.quantumlab.de) Quantumrandom number generator with single photons asinteractive experiment.Random.org (http://www.random.org) generates randomnumbers using atmospheric noises (see also Random.org).HotBits (http://www.fourmilab.ch/hotbits/) generates random numbers from radioactive decay.QRBG (http://random.irb.hr) Quantum Random Bit GeneratorQRNG (http://qrng.physik.hu­berlin.de/) Fast Quantum Random Bit GeneratorChaitin: Randomness and Mathematical Proof(http://www.cs.auckland.ac.nz/CDMTCS/chaitin/sciamer.html)A Pseudorandom Number Sequence Test Program (Public Domain)(http://www.fourmilab.ch/random/)Dictionary of the History of Ideas: (http://etext.lib.virginia.edu/cgi­local/DHI/dhi.cgi?id=dv1­46)Chance

13. ^ a b Johnson, George (8 June 2008). "Playing the Odds"(http://www.nytimes.com/2008/06/08/books/review/Johnson­G­t.html?_r=1). The New York Times.

12/12/2014 Randomness ­ Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Randomness 11/11

Philosophy: Free Will vs. Determinism (http://www.spaceandmotion.com/Philosophy­Free­Will­Determinism.htm)RAHM Nation Institute (http://www.rahmnation.org)History of randomness definitions (http://www.wolframscience.com/nksonline/page­1067b­text),in Stephen Wolfram's A New Kind of ScienceComputing a Glimpse of Randomness(http://www.cs.auckland.ac.nz/~cristian/Calude361_370.pdf)Chance versus Randomness (http://plato.stanford.edu/entries/chance­randomness/), from theStanford Encyclopedia of Philosophy

Retrieved from "http://en.wikipedia.org/w/index.php?title=Randomness&oldid=628417419"

Categories: Cryptography Statistical randomness Randomness

This page was last modified on 6 October 2014 at 01:54.Text is available under the Creative Commons Attribution­ShareAlike License; additional termsmay apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is aregistered trademark of the Wikimedia Foundation, Inc., a non­profit organization.