Randomness

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When most people think of randomness, they generally think of a condition with an apparent absence of a regular plan, pattern, or purpose. The word random is derived from the Old French word randon, meaning haphazard. The mathematical meaning is not significantly different from the common usage. Mathematical randomness is exhibited when the next state of a process cannot be exactly determined from the previous state. Randomness involves uncertainty. The most common example of randomness is the tossing of a coin. From the result of a previous toss, one cannot predict with certainty that the result of the next coin toss will be either heads or tails.

People performing statistical studies or requiring random numbers for other applications obtain them from a table, calculator, or computer. Random digits can be generated by repeatedly selecting from the set of numbers {0, 1, 2,…, 9}. One way of making the selection would be to number ten balls with these digits and then draw one ball at a time without looking, recording the number on the drawn ball and replacing the ball after each successive drawing. The recorded string of digits would be a set of random numbers. Extensive tables of random numbers have been generated in the past. For example, the RAND Corporation used an electronic roulette wheel to compile a book with a million random digits.

Today, rather than using tables, people requiring random numbers more frequently use a calculator or a computer. Computers and calculators have programs that generate random numbers, but the numbers are really not random because they are based on complicated, but nonetheless deterministic, computational algorithms. These algorithms generate a sequence of what are called pseudo-random numbers.

Using computers to generate random numbers has altered the definition of randomness to involve the complexity of the algorithm used in the computations. It is not possible to achieve true randomness with a computer because there is always some underlying process that, with tremendous computational difficulty, could be duplicated to replicate the pseudo-random number. Physicists consider the emissions from atoms to be a truly random process, and therefore a source of generating random numbers. But the instruments used to detect the emissions introduce limitations on the actual randomness of numbers produced in that process. So at best, the many ways of generating a random number only approximate true randomness. With the advent of computers, mathematicians can define and develop methods to measure the randomness of a given number, but have yet to prove that a number sequence is truly random.

Randomness has very important applications in many areas of mathematics. In statistics, the selection of a random sample is important to ensure that a study is conducted without bias. A simple random sample is obtained by numbering every member of the population of interest, and assigning each member a numerical label. The appropriate sample size is determined. The researcher then obtains the same quantity of random numbers as the sample size from a table of random numbers, a calculator, or a computer. The members of the population labeled with the corresponding random numbers are selected for study. In this way, every member of the population has an equal likelihood of being selected, eliminating any bias that may be introduced by other selection methods. Ensuring the randomness of the selection makes the results of the study more scientifically valid and more likely to be replicated.

There are many other applications of randomness in mathematics. Using solution methods involving random walks, applied mathematicians can obtain solutions for complex mathematical models that are the basis of modern physics. Albert Einstein, and later Norbert Weiner, used the method in the early twentieth century to describe the motion of microscopic particles suspended in a fluid. In the late 1940s, mathematicians Stanislaw Ulam and John von Neumann developed Monte Carlo methods, which apply random numbers to solve deterministic models arising in nuclear physics and engineering. Randomness is also important in the mathematics of cryptography, which is particularly important today and will continue to be in the future as sensitive information is transmitted across the Internet. Seemingly random numbers are used as the keys to encryption systems in use in digital communications.

In more complicated examples, randomness is closely tied to probability. Even seemingly irregular random phenomena exhibit some long-term regularity. Probability theory mathematically explains randomness. Mathematicians sometimes divide processes they study into deterministic or probabilistic (or stochastic) models. If a phenomenon can be modeled deterministically, the process can be predicted with certainty using mathematical formulas and relationships. Stochastic models involve uncertainty, but with probability theory, the uncertain behavior of the phenomenon is better understood despite the haphazardness. One cannot predict the specific outcome of the coin-tossing experiment, but you can achieve an expectation and understanding of the process using probability theory. Through the use of probability theory, one understands much about topics such as nuclear physics.

Not all processes can be classified as deterministic or stochastic in an obvious manner. Chaos theory is a relatively recent area of mathematical study that helps explain the randomness that appears in some processes that are otherwise considered to be deterministic. The behavior of chaotic systems is dramatically influenced by their sensitivity to small changes in initial conditions. Mathematicians are currently developing methods to understand the underlying order of chaotic systems. Mathematicians apply chaos theory to clarify the apparent randomness of some processes.

Chaos; Cryptology.

Abramowitz, Milton, and Irene Stegun, eds. Handbook of Mathematical Functions. Washington, D.C.: U.S. Department of Commerce, National Bureau of Standards, 1964.

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Peterson, Ivars. The Jungles of Randomness. New York: John Wiley & Sons, 1998.

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