Randomness
Randomness is a concept that refers to the occurrence of events without a predictable pattern or cause, challenging deterministic views that assert every event is traceable to a specific origin. Historically, many cultures have grappled with the ideas of fate and chance, as exemplified by ancient gambling practices and philosophical inquiries, such as Aristotle's categorization of events. The development of probability theory in the 17th century established a mathematical framework for understanding randomness, with figures like Blaise Pascal and Pierre de Fermat laying foundational concepts that continue to evolve. In modern contexts, randomness is crucial across various applications, from gambling and encryption to statistical sampling and natural phenomenon modeling.
Generating randomness, particularly in computational settings, differs from merely observing it. While mechanical methods like dice have historically been used, they often introduce biases, prompting the development of pseudorandom number generators and hardware-based solutions that aim for true randomness. Researchers continually refine techniques to assess and improve randomness through various tests, acknowledging its counterintuitive nature. Beyond classical views of probability, concepts such as entropy from information theory and Bayesian statistics have expanded the understanding of randomness, influencing fields as diverse as data science and risk assessment. This multifaceted nature of randomness highlights its significance in both theoretical and practical realms.
Randomness
Summary: While a seemingly simple idea, the concept of randomness has been studied by mathematicians for thousands of years and has many modern applications.
The philosophical concept of determinism supposes that all events that occur in the world can be traced back to a specific precipitating cause and denies the possibility that chance may influence predestined causal paths. Mathematical determinism similarly states that, given initial conditions and a mathematical function or system, there is only one possible outcome no matter how many times the calculation is performed.
Historical Studies of Randomness and Certainty
Many ancient cultures embraced the idea of fate. For example, the Greek pantheon included goddesses known as Fates. At the same time, the existence of ancient gambling games and deities like the Roman goddess Fortuna suggest that these people understood the notion of randomness or chance on some level. Around 300 b.c.e., Aristotle proposed dividing events into three different categories: certain events, which were deterministic; probable events, which were because of chance; and unknowable events.
In the 1600s, the work of mathematicians such as Blaise Pascal and Pierre de Fermat laid some foundations for modern probability theory, which quantifies chance. Abraham de Moivre published The Doctrine of Chances in 1718. Around the same time, Daniel Bernoulli investigated randomness in his Exposition of a New Theory on the Measurement of Risk. Nonetheless, determinism continued to maintain a prominent place in mathematics and science. Researchers often assumed that seemingly observed randomness in their data was because of measuring error or a lack of complete understanding of the phenomena being observed.
The emergence of fields like statistics and quantum mechanics in the nineteenth century helped drive new work on randomness. Mathematician Émile Borel wrote more than 50 papers on the calculus of probability between 1905 and 1950, emphasizing the diverse ways in which randomness could be applied in the natural and social sciences as well as in mathematics. Applied probabilistic modeling grew very quickly after World War II.
Randomness in Society
In twenty-first-century colloquial speech, the word “random” is often used to mean events that cannot be predicted, similar to Aristotle’s unknowable classification. However, probability theory can model the long-term behavior of random or stochastic systems using probability distribution functions, which are essentially sets of possible outcomes having mathematically definable probabilities of occurring. They describe the overall relative frequencies of events or ranges of events, though the specific sequence of individual events cannot be completely determined. Stochastic behavior is observed in many natural systems, such as atmospheric radiation, consumer behavior, the variation of characteristics in biological systems, and the stock market. It is also connected to mathematical concepts like logarithms and the digits of π. Elementary school children discuss some of the basics of randomness when studying data collection methods, like surveys and experiments. Formal mathematical explorations typically begin in high school and continue through college.
Society depends on the use of randomness or the assumption that randomness is involved in a given process. Examples include operating gambling games and lotteries; encrypting coded satellite transmissions; securing credit card data for e-transactions; allocating drugs in experimental trials; sampling people in surveys; establishing insurance rates; creating key patterns for locks; and modeling complex natural phenomena such as weather and the motion of subatomic particles.
Generating Randomness
Generating random numbers, however, is very different from observing random behavior. For example, in 1995, graduate students Ian Goldberg with David Wagner discovered a serious flaw in the system used to generate temporary random security keys in the Netscape Navigator Web browser. Almost every civilization in recorded history has used mechanical systems, such as dice, for generating random numbers and randomness has close ties with gaming and game theory. Physical methods are not generally practical for quickly generating the large sequences of random numbers needed for Monte Carlo simulation and other computational techniques. Flaws in shuffling and physical characteristics, like a worn-down corner on a die, or deliberate human intervention, can also introduce bias. In fact, some people have proven their ability to flip a coin in a predetermined pattern. Motivated by the mathematical unreliability of these physical systems, mathematicians and scientists sought other reliable sources of randomness. Leonard Tippet used census data, believed to be random, to create a table of 40,000 random digits in 1927. Ronald Fisher used the digits of logarithms to generate additional random tables in 1938. In 1955, RAND Corporation published A Million Random Digits with 100,000 Normal Deviates, which were generated by an electronic roulette wheel. Random digit tables are still routinely used by researchers who need to perform limited tasks like randomizing subjects to treatment groups in experimental designs as well as in many statistics classes.
The development of computers in the middle of the twentieth century allowed mathematicians, such as John von Neumann, and computer scientists to generate “pseudorandom” numbers. The name comes from the fact that the digits are produced by some type of deterministic mathematical algorithm that will eventually repeat in a cycle, though relatively shorter runs will display characteristics similar to truly random numbers. Using very large numbers, or trigonometric or logarithmic functions, tends to create longer non-repeating sequences. Linear feedback shift registers are frequently used for applications such as signal broadcast and stream cyphers. Linear congruential generators produce numbers that are more likely to be serially correlated, but they are useful in applications like video games, where true randomness is not as critical and many random streams are needed at the same time. Hardware random number generators, built as an alternative to algorithm-driven software generators, are based on input from naturally occurring phenomena like radioactive decay or atmospheric white noise and produce what their creators believe to be truly random numbers.
Randomness Tests
Mathematicians and computer scientists are perpetually working on methods to improve pseudorandom number algorithms and to determine whether observed data are truly random. Randomness can be counterintuitive. For example, the sequences 6, 6, 6, 6, 6, 6 and 2, 6, 1, 5, 5, 4 produced by fair rolls of a six-sided die are equally likely to occur, but most people would say that the first sequence does not “look” random. Irregularity and the absence of obvious patterns are useful ideas, but they are difficult to measure. Distinctions between local and global regularity must also be made, which include the ideas of finite sets and infinite sets. Irenée-Jules Bienaymé proposed a simple test for randomness of observations on a continuously varying quantity in the nineteenth century. Florence Nightingale David published a power function for randomness tests shortly after World War II. Another technique from information theory measures randomness for a given sequence by calculating the shortest Turing machine program that could produce the sequence. The National Institute of Standards and Technology recommends many such tests, including binary matrix rank, discrete Fourier transform, linear complexity, and cumulative or overlapping sums. As of 2010, the digits of π had passed all commonly used randomness tests.
Classical probability theory is not the only way to think about randomness. Claude Shannon’s development of information theory in the 1940s resulted in the entropy view of randomness, which is now widely used in many scientific fields. By the latter half of the twentieth century, fuzzy logic and chaos theory also emerged. Fuzzy logic was initially derived from Lotfali Zadeh’s work on fuzzy sets and non-binary truth values, while chaos theory dates back to Henri Poincaré’s explorations of the three body problem. Bayesian statistics, based on the eighteenth-century work of Thomas Bayes, challenges the frequentist approach by allowing randomness to be conceptualized and quantified as a partial belief, which shares characteristics with fuzzy logic. Spam filtering is one application that relies on Bayesian notions of randomness.
Bibliography
Bennett, Deborah. Randomness. Cambridge, MA: Harvard University Press, 1998.
Mlodinow, Leonard. The Drunkard’s Walk: How Randomness Rules Our Lives. New York: Pantheon Books, 2008.
Random.org. http://www.random.org.