Randomness
Part of a series on statistics |
Probability theory |
---|
In common usage, randomness is the apparent or actual lack of definite
The fields of mathematics, probability, and statistics use formal definitions of randomness, typically assuming that there is some 'objective' probability distribution. In statistics, a
Randomness is most often used in
.Random selection, when narrowly associated with a simple random sample, is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. A random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, say research subjects, has the same probability of being chosen, then we can say the selection process is random.[2]
According to Ramsey theory, pure randomness (in the sense of there being no discernible pattern) is impossible, especially for large structures. Mathematician Theodore Motzkin suggested that "while disorder is more probable in general, complete disorder is impossible".[4] Misunderstanding this can lead to numerous conspiracy theories.[5] Cristian S. Calude stated that "given the impossibility of true randomness, the effort is directed towards studying degrees of randomness".[6] It can be proven that there is infinite hierarchy (in terms of quality or strength) of forms of randomness.[6]
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 to attempt to circumvent randomness and fate.[7][8] Beyond religion and games of chance, randomness has been attested for sortition since at least ancient Athenian democracy in the form of a kleroterion.[9]
The formalization of odds and chance was perhaps earliest done by the Chinese of 3,000 years ago. The Greek philosophers discussed randomness at length, but only in non-quantitative forms. It was only in the 16th century that Italian mathematicians began to formalize the odds associated with various games of chance. The invention of calculus had a positive impact on the formal study of randomness. In the 1888 edition of his book The Logic of Chance, John Venn wrote a chapter on The conception of randomness that included his view of the randomness of the digits of pi (π), by using them to construct a random walk in two dimensions.[10]
The early part of the 20th century saw a rapid growth in the formal analysis of randomness, as various approaches to the mathematical foundations of probability were introduced. In the mid-to-late-20th century, ideas of
Although randomness had often been viewed as an obstacle and a nuisance for many centuries, in the 20th 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
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 of statistical mechanics to explain phenomena in thermodynamics and the properties of gases.
According to several standard interpretations of
In biology
The modern evolutionary synthesis ascribes the observed diversity of life to random genetic mutations followed by natural selection. The latter retains some random mutations in the gene pool due to the systematically improved chance for survival and reproduction that those mutated genes confer on individuals who possess them. The location of the mutation is not entirely random however as e.g. biologically important regions may be more protected from mutations.[14][15][16]
Several authors also claim that evolution (and sometimes development) requires a specific form of randomness, namely the introduction of qualitatively new behaviors. Instead of the choice of one possibility among several pre-given ones, this randomness corresponds to the formation of new possibilities.[17][18]
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
As far as behavior is concerned, 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 arose from attempts to formulate mathematical descriptions of chance events, originally in the context of gambling, but later in connection with physics. Statistics is used to infer an 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.
Randomness occurs 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 considered likely to be normal:
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.[22]
In statistics
In statistics, randomness is commonly used to create
In information science
In information science, irrelevant or meaningless data is considered noise. Noise consists of numerous 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 terms of the development of random networks, for communication randomness rests on the two simple assumptions of Paul Erdős and Alfréd Rényi, who said that there were a fixed number of nodes and this number remained fixed for the life of the network, and that all nodes were equal and linked randomly to each other.[clarification needed][23]
In finance
The random walk hypothesis considers that asset prices in an organized market evolve at random, in the sense that the expected value of their change is zero but the actual value may turn out to be positive or negative. More generally, asset prices are influenced by a variety of unpredictable events in the general economic environment.
In politics
Random selection can be an official method to resolve
Randomness and religion
Randomness can be seen as conflicting with the
In some religious contexts, procedures that are commonly perceived as randomizers are used for divination. Cleromancy uses the casting of bones or dice to reveal what is seen as the will of the gods.
Applications
In most of its mathematical, political, social and religious uses, randomness is used for its innate "fairness" and lack of bias.
Politics:
Games: Random numbers were first investigated in the context of
Sports: Some sports, including
Mathematics: Random numbers are also employed 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.,
Religion: 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 for more).
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 and dice).
- Randomness intrinsically generated by the system. This is also called arithmetics or cellular automaton) for generating pseudorandom numbers. The behavior of the system can be determined by knowing the seed stateand 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 they can generate random numbers.
Before the advent of computational
Measures and tests
There are many practical measures of randomness for a binary sequence. These include measures based on frequency, discrete transforms, complexity, or a mixture of these, such as the tests by Kak, Phillips, Yuen, Hopkins, Beth and Dai, Mund, and Marsaglia and Zaman.[26]
Quantum nonlocality has been used to certify the presence of genuine or strong form of randomness in a given string of numbers.[27]
Misconceptions and logical fallacies
Popular perceptions of randomness are frequently mistaken, and are often based on fallacious reasoning or intuitions.
Fallacy: a number is "due"
This argument is, "In a random selection of numbers, since all numbers eventually appear, those that have not come up yet are 'due', and thus more likely to come up soon." This logic is only correct if applied to a system where numbers that come up are removed from the system, such as when playing cards are drawn and not returned to the deck. In this case, once a jack is removed from the deck, the next draw is less likely to be a jack and more likely to be some other card. However, if the jack is returned to the deck, and the deck is thoroughly reshuffled, a jack is as likely to be drawn as any other card. The same applies in any other process where objects are selected independently, and none are removed after each event, such as the roll of a die, a coin toss, or most lottery number selection schemes. Truly random processes such as these do not have memory, which makes it impossible for past outcomes to affect future outcomes. In fact, there is no finite number of trials that can guarantee a success.
Fallacy: a number is "cursed" or "blessed"
In a random sequence of numbers, a number may be said to be cursed because it has come up less often in the past, and so it is thought that it will occur less often in the future. A number may be assumed to be blessed because it has occurred more often than others in the past, and so it is thought likely to come up more often in the future. This logic is valid only if the randomisation might be biased, for example if a die is suspected to be loaded then its failure to roll enough sixes would be evidence of that loading. If the die is known to be fair, then previous rolls can give no indication of future events.
In nature, events rarely occur with a frequency that is known
Fallacy: odds are never dynamic
In the beginning of a scenario, one might calculate the probability of a certain event. However, as soon as one gains more information about the scenario, one may need to re-calculate the probability accordingly.
For example, when being told that a woman has two children, one might be interested in knowing if either of them is a girl, and if yes, what is probability that the other child is also a girl. Considering the two events independently, one might expect that the probability that the other child is female is ½ (50%), but by building a probability space illustrating all possible outcomes, one would notice that the probability is actually only ⅓ (33%).
To be sure, the probability space does illustrate four ways of having these two children: boy-boy, girl-boy, boy-girl, and girl-girl. But once it is known that at least one of the children is female, this rules out the boy-boy scenario, leaving only three ways of having the two children: boy-girl, girl-boy, girl-girl. From this, it can be seen only ⅓ of these scenarios would have the other child also be a girl[28] (see Boy or girl paradox for more).
In general, by using a probability space, one is less likely to miss out on possible scenarios, or to neglect the importance of new information. This technique can be used to provide insights in other situations such as the Monty Hall problem, a game show scenario in which a car is hidden behind one of three doors, and two goats are hidden as booby prizes behind the others. Once the contestant has chosen a door, the host opens one of the remaining doors to reveal a goat, eliminating that door as an option. With only two doors left (one with the car, the other with another goat), the player must decide to either keep their decision, or to switch and select the other door. Intuitively, one might think the player is choosing between two doors with equal probability, and that the opportunity to choose another door makes no difference. However, an analysis of the probability spaces would reveal that the contestant has received new information, and that changing to the other door would increase their chances of winning.[28]
See also
- Chaitin's constant
- Chance (disambiguation)
- Frequentist probability
- Indeterminism
- Nonlinear system
- Probability interpretations
- Probability theory
- Pseudorandomness
- Random.org—generates random numbers using atmospheric noise
- Sortition
Notes
- ^ Strictly speaking, the frequency of an outcome will converge almost surely to a predictable value as the number of trials becomes arbitrarily large. Non-convergence or convergence to a different value is possible, but has probability zero. Consistent non-convergence is thus evidence of the lack of a fixed probability distribution, as in many evolutionary processes.
References
- ^ The Oxford English Dictionary defines "random" as "Having no definite aim or purpose; not sent or guided in a particular direction; made, done, occurring, etc., without method or conscious choice; haphazard."
- ^ a b "Definition of randomness | Dictionary.com". www.dictionary.com. Retrieved 21 November 2019.
- ^ Third Workshop on Monte Carlo Methods, Jun Liu, Professor of Statistics, Harvard University
- S2CID 37243306.
- ^ Ted.com, (May 2016). The origin of countless conspiracy theories
- ^ a b Cristian S. Calude, (2017). "Quantum Randomness: From Practice to Theory and Back" in "The Incomputable Journeys Beyond the Turing Barrier" Editors: S. Barry Cooper, Mariya I. Soskova, 169–181, doi:10.1007/978-3-319-43669-2_11.
- ISBN 0-19-512332-8page 279
- ISBN 0-674-01517-7page 370
- ISBN 9780631180173.
- ISBN 0-387-98844-0page 115. The 1866 edition of Venn's book (on Google books) does not include this chapter.
- ^ Reinert, Knut (2010). "Concept: Types of algorithms" (PDF). Freie Universität Berlin. Retrieved 20 November 2019.
- S2CID 4412358.
- ^ "Each nucleus decays spontaneously, at random, in accordance with the blind workings of chance." Q for Quantum, John Gribbin
- U.C. Davis. Retrieved 12 February 2022.
- PMID 35022609.
- PMID 29233924.
- S2CID 15609415.
- S2CID 55589891.
- S2CID 72016377.
The distribution of freckles seems entirely random, and not associated with any other obviously punctuate anatomical or physiological feature of skin.
- .
- ^ Yongge Wang: Randomness and Complexity. PhD Thesis, 1996. http://webpages.uncc.edu/yonwang/papers/thesis.pdf
- ^ "Are the digits of pi random? researcher may hold the key". Lbl.gov. 23 July 2001. Archived from the original on 20 October 2007. Retrieved 27 July 2012.
- ^ Laszso Barabasi, (2003), Linked, Rich Gets Richer, P81
- ^ 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 of votes, the clerk shall choose the successful candidate or candidates by lot."
- ISBN 978-1-349-11899-1.
- ^ Terry Ritter, Randomness tests: a literature survey. ciphersbyritter.com
- S2CID 4300790.
- ^ a b Johnson, George (8 June 2008). "Playing the Odds". The New York Times.
Further reading
- Randomness by ISBN 0-674-10745-4.
- Random Measures, 4th ed. by MR0854102.
- The Art of Computer Programming. Vol. 2: Seminumerical Algorithms, 3rd ed. by ISBN 0-201-89684-2.
- ISBN 1-58799-190-X.
- Exploring Randomness by ISBN 1-85233-417-7.
- Random by Kenneth Chan includes a "Random Scale" for grading the level of randomness.
- The Drunkard’s Walk: How Randomness Rules our Lives by ISBN 978-0-375-42404-5.
External links
- QuantumLab Quantum random number generator with single photons as interactive experiment.
- HotBits generates random numbers from radioactive decay.
- QRBG Quantum Random Bit Generator
- QRNG Fast Quantum Random Bit Generator
- Chaitin: Randomness and Mathematical Proof
- A Pseudorandom Number Sequence Test Program (Public Domain)
- Dictionary of the History of Ideas: Chance
- Computing a Glimpse of Randomness
- Chance versus Randomness, from the Stanford Encyclopedia of Philosophy