Randomness: Difference between revisions

[[File:RandomBitmap.png|thumb|alt=|A [[Pseudorandom Number Generator|pseudorandomly generated]] [[bitmap]]. ]]

In common parlanceusage, '''randomness''' is the apparent or actual lack of definite [[pattern]] or [[predictability]] in eventsinformation.<ref>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."</ref><ref name=":0">{{Cite web|url=https://www.dictionary.com/browse/randomness|title=Definition of randomness {{!}} Dictionary.com|website=www.dictionary.com|language=en|access-date=2019-11-21}}</ref> A random sequence of events, [[symbol]]s or steps often has no [[:wikt:order|order]] and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if thethere is a known [[probability distribution]] is known, the frequency of different outcomes over repeated events (or "trials") is predictable.<ref group="note">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.</ref> For example, when throwing two [[dice]], the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, [[probability]], and [[information entropy]].

The fields of mathematics, probability, and statistics use formal definitions of randomness, typically assuming that there is some 'objective' probability distribution. In statistics, a [[random variable]] is an assignment of a numerical value to each possible outcome of an [[event space]]. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in [[random sequence]]s. A [[random process]] is a sequence of random variables whose outcomes do not follow a [[determinism|deterministic]] pattern, but follow an evolution described by [[probability distribution]]s. These and other constructs are extremely useful in [[probability theory]] and the various [[applications of randomness]].

'''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. Note that aA 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.<ref name=":0" />

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".<ref>{{cite journal|author=Hans Jürgen Prömel|year=2005|title=Complete Disorder is Impossible: The Mathematical Work of Walter Deuber|journal=[[Combinatorics, Probability and Computing]]|publisher=Cambridge University Press|volume=14|pages=3–16|doi=10.1017/S0963548304006674|s2cid=37243306|url=http://edoc.hu-berlin.de/18452/28352 }}</ref> Misunderstanding this can lead to numerous [[conspiracy theory|conspiracy theories]].<ref>Ted.com, (May 2016). [https://www.ted.com/talks/patrickjmt_the_origin_of_countless_conspiracy_theories/transcript The origin of countless conspiracy theories]</ref> [[Cristian S. Calude]] stated that "given the impossibility of true randomness, the effort is directed towards studying degrees of randomness".<ref name="calude2017">[[Cristian S. Calude]], (2017). [https://www.cs.auckland.ac.nz/~cristian/crispapers/QR_TP.pdf "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.</ref> It can be proven that there is infinite hierarchy (in terms of quality or strength) of forms of randomness.<ref name="calude2017" />

[[File:Pompeii - Osteria della Via di Mercurio - Dice Players.jpg|thumb|Ancient [[fresco]] of dice players in [[Pompei]].]]

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.<ref>''Handbook to life in ancient Rome'' by Lesley Adkins 1998 {{isbn|0-19-512332-8}} page 279</ref><ref>''Religions of the ancient world'' by Sarah Iles Johnston 2004 {{isbn|0-674-01517-7}} page 370</ref> Beyond [[religion]] and [[game of chance|games of chance]], randomness has been attested for [[sortition]] since at least ancient [[Athenian democracy]] in the form of a [[kleroterion]].<ref name=Hansen>{{cite book

The Chineseformalization of 3000odds yearsand agochance werewas perhaps the earliest peopledone toby formalizethe oddsChinese andof chance3,000 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.<ref>''Annotated readings in the history of statistics'' by Herbert Aron David, 2001 {{isbn|0-387-98844-0}} page 115. Note that theThe 1866 edition of Venn's book (on Google books) does not include this chapter.</ref>

The [[modern synthesis (20th century)|modern evolutionary synthesis]] ascribes the observed diversity of life to random genetic [[mutation]]s 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.<ref>{{cite news |title=Study challenges evolutionary theory that DNA mutations are random |url=https://phys.org/news/2022-01-evolutionary-theory-dna-mutations-random.html |access-date=12 February 2022 |work=[[U.C. Davis]] |language=en}}</ref><ref>{{cite journal |last1=Monroe |first1=J. Grey |last2=Srikant |first2=Thanvi |last3=Carbonell-Bejerano |first3=Pablo |last4=Becker |first4=Claude |last5=Lensink |first5=Mariele |last6=Exposito-Alonso |first6=Moises |last7=Klein |first7=Marie |last8=Hildebrandt |first8=Julia |last9=Neumann |first9=Manuela |last10=Kliebenstein |first10=Daniel |last11=Weng |first11=Mao-Lun |last12=Imbert |first12=Eric |last13=Ågren |first13=Jon |last14=Rutter |first14=Matthew T. |last15=Fenster |first15=Charles B. |last16=Weigel |first16=Detlef |title=Mutation bias reflects natural selection in Arabidopsis thaliana |journal=Nature |date=February 2022 |volume=602 |issue=7895 |pages=101–105 |doi=10.1038/s41586-021-04269-6 |pmid=35022609 |pmc=8810380 |bibcode=2022Natur.602..101M |language=en |issn=1476-4687}}</ref><ref>{{cite journal |last1=Belfield |first1=Eric J. |last2=Ding |first2=Zhong Jie |last3=Jamieson |first3=Fiona J.C. |last4=Visscher |first4=Anne M. |last5=Zheng |first5=Shao Jian |last6=Mithani |first6=Aziz |last7=Harberd |first7=Nicholas P. |title=DNA mismatch repair preferentially protects genes from mutation |journal=Genome Research |date=January 2018 |volume=28 |issue=1 |pages=66–74 |doi=10.1101/gr.219303.116|pmid=29233924 |pmc=5749183 }}</ref>

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.<ref>{{Cite book|last1=Longo|first1=Giuseppe|last2=Montévil|first2=Maël|last3=Kauffman|first3=Stuart|datetitle=2012-01-01Proceedings of the 14th annual conference companion on Genetic and evolutionary computation |titlechapter=No Entailingentailing Lawslaws, but Enablementenablement in the Evolutionevolution of the Biospherebiosphere |date=2012-01-01|url=https://www.academia.edu/11720588|journal=Proceedings of the 14th Annual Conference Companion on Genetic and Evolutionary Computation|series=GECCO '12|location=New York, NY, USAUS|publisher=ACM|pages=1379–1392|doi=10.1145/2330784.2330946|isbn=9781450311786|arxiv=1201.2069|citeseerx=10.1.1.701.3838|s2cid=15609415}}</ref><ref>{{Cite journal|last1=Longo|first1=Giuseppe|last2=Montévil|first2=Maël|date=2013-10-01|title=Extended criticality, phase spaces and enablement in biology|url=https://www.academia.edu/11720575|journal=Chaos, Solitons & Fractals|series=Emergent Critical Brain Dynamics|volume=55|pages=64–79|doi=10.1016/j.chaos.2013.03.008|bibcode=2013CSF....55...64L|s2cid=55589891 }}</ref>

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 thean underlying [[probability distribution]] of a collection of empirical observations. For the purposes of [[simulation]], it is necessary to have a large supply of [[Random sequence|random numbers]]—or means to generate them on demand.

|text = 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.<ref>{{cite web |url=http://www.lbl.gov/Science-Articles/Archive/pi-random.html |title=Are the digits of pi random? researcher may hold the key |publisher=Lbl.gov |date=2001-07-23 |access-date=2012-07-27 |archive-date=20 October 2007 |archive-url=https://web.archive.org/web/20071020010208/http://lbl.gov/Science-Articles/Archive/pi-random.html |url-status=dead }}</ref>

Random selection can be an official method to resolve [[Tie (draw)|tied]] elections in some jurisdictions.<ref>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."</ref> Its use in politics originates long ago. Many offices in [[Ancientancient Athens]] were chosen by lot instead of modern voting.

'''Sports''': Some sports, including [[American football]], use [[coin toss]]es to randomly select starting conditions for games or [[seed (sports)|seed]] tied teams for [[playoffs|postseason play]]. The [[National Basketball Association]] uses a weighted [[NBA Draftdraft Lotterylottery|lottery]] to order teams in its draft.

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 isthe 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%).

* [[FrequencyFrequentist probability]]