What Is Random?

Kevin McKeen whom I quoted elsewhere (In a world as crazy as this one, it ought to be easy to find something that happens solely by chance. It isn't.) explains the process (that was supposed to be random) of the 1969 draft lottery (The Orderly Pursuit of Pure Disorder, Discover, January, 1981):

Capsules containing all the January birth dates were put in a box and mixed up; then all the February dates; then all the March dates, and so forth. As a result, dates that fell late in the year didn't get as thoroughly stirred up as those from early in the year. And the night the capsules were drawn, birthdays late in the year turned up first: a December date had a better than even chance of being among the first third selected. The next year draft officials mixed the dates better, and the problem was resolved.

Not surprisingly, many reluctant draftees questioned the fairness of the procedure. Paul J. Campbell (Mathematics Magazine, 84 (2011), p. 393) found that

The specifications for the Vietnam-War-era draft lotteries and for the current annual diversity immigration lottery include the word "random." The meaning has been the subject of court cases: Is "random" supposed to refer to the process of selection or to the results, or both? Does it require equal probabilities for individuals? Is independence required? The courts seem to be looking for a combination of fairness and equality.

Campbell gave three examples where judges struggled with the notion of randomness:

  • There must be "no plan, purpose or pattern in the drawing . . . perfectly fair . . . no discrimination . . . no inequity" (U.S. v. Kotrlik, et al., 465 F.2d 976-977, re the 1970 draft lottery);

  • Equal probability is to be "approached as closely as reasonably possible under all the circumstances" (Stodolsky v. Hershey, 2 Selective Serv. L. Rptr. 3527, 3528-29 (W.D. Wis. 1969);

  • The process should embody "qualities of unpredictability and equal probability" (Smirnov, et al. v. Clinton, et al., United States District Court, District of Columbia, July 14, 2011, Civil Action No. 11-1126 (ABJ).)

The Cambridge Dictionary of Statistics defines random as Governed by chance; not completely determined by other factors. Non-deterministic.

Pierre-Simon de Laplace (1749 - 1827) was of the notion that

... the word chance expresses nothing more than our lack of knowledge about the causes of events that we see appear and follow one another in an order that is invisible to us.

But in 1927 Weiner Heisenberg announced the Uncertainty Principle that ultimately meant the objective presence of randomness. In his words:

In the precise formulation of the law of causality - "If we know the present, we can calculate the future" - it is not the conclusion that is false, but the hypothesis.

Statistics is more concerned with the concept of randomness than the branch of pure mathematics known as Probability Theory. Statistics is preoccupied with making predictions, e.g., those that are related to consumer behavior or public political preferences. To make reliable predictions statisticians simply have to insure randomness of the population sample. For a mathematician, on the other hand, the nature of randomness is less essential than the mere fact of its existence. So in mathematics, one may encounter a definition like that in [ Ash, p.49] - A random variable is a real-valued function on a sample space - that does not go beyond terminological reminder of the origins.

A. N. Kolmogorov - the father of the modern probabilistic formalism - concurs [Kolmogorov, p. 244]:

A random variable is the name of a quantity which under given conditions S may take various values with specific probabilities.

Before defining the concept of probability [Kolmogorov, p. 229] observes:

An event A, which under a complex of conditions S sometimes occur and sometimes does not occur, is called random with respect to the complex of conditions. This raises the question: Does the randomness of the event A demonstrate the absence of any law connecting the complex of conditions S and the event A?

The question is of cause rhetorical, and the answer is No. Discovering these laws is the subject of Probability theory.

[Darling] highlights three possible aspects of "random": Without cause; not compressible; obeying the statistics of a fair coin toss. The first and last are rather questionable. The first characterization may be objected on the grounds that some random variables depend on each other so that the values of each are not exactly without cause. Quantum mechanics, also, did not make away with the idea of causality but reinterpreted it. The fact that various random variables can be distributed differently weakens the third characterization. But the second aspect (not compressible) is an elegant characteristics of what is known as "algorithmic randomness."

Generation of random sequences of numbers and the question of randomness of their digits are of great importance in computer science and its applications. [Bewersdorff, p. 41]: An important characteristic of randomness is that its results should be unpredictable. Thus the sequence of decimal digits of the number π are not in fact random. π is an infinite decimal but there are (finite) algorithms for computing its digits - however distant. Numbers whose digits can be computed algorithmically are called computable; π is one such number. All rational numbers are also computable (all by the same algorithm - long division - but with different input). No computable number may be said to have random digits. But some - and these are called pseudorandom - appear to have a more random-like distribution of digits than others. For example, this is true of π compared to any rational number with a short decimal period.

If a number is thought as carrying some information, then the algorithm for computing a number may be said to carry exactly same information, but being finite, the algorithm could be presented with fewer symbols than an infinite decimal. Such algorithms compress (but do not distort) the information carried by the numbers. The length - or rather shortness - of the algorithms is an indication of how distant the corresponding numbers are from being pseudorandom. A truly random number may perhaps be expressed with an infinite algorithm, if such are admitted.

This concept of randomness was developed in the 1960s by A. N. Kolmogorov and, independently, G. J. Chaitin. Chaitin came up with uncomputable number Ω of which he proudly says [Chaitin, p. 19],

You see, I've shown that there's complete randomness, no pattern, lack of structure, and that reasoning is completely useless, if you're interested in the individual bits of the number Ω.

In line with my earlier remark, [Bewersdorff, p. 46] observes that As interesting as such characterization of randomness may be, it holds, surprisingly, hardly any significance for mathematical probability theory. A mathematical model of probability can be constructed without reference for randomness.

Interestingly, the concept of randomness showed unexpectedly in mathematics forcing us to reconsider the ages old concept of determinism and its relation to randomness, or the lack thereof. The modern idea of chaos is intricately related to the deterministic dynamic systems.

Here's what [Stewart, p. 317] says following the description of the now common period doubling process:

What do I mean by "random" here? It's a tricky question to answer, because the very phenomenon just discussed shows that it is not as simple an idea as as might be imagined. "Random" means that no obvious structure exists, but that "on average" we can say various things, such as how often the values occur in a given range. But in the past, "random" has carried the connotation of "indeterminate", that is, a system is deterministic if it follows exactly some regular law, random if not.

However, here we found a deterministic system (it obeys the formula xn+1 = kxn(1 - xn) exactly) whose behavior is more sensibly described as random!

Etymologically, the word "random" did not originate with the ancient Greeks nor Romans. According to [The Words of Mathematics],

random (adjective): from Old French randir "to run impetuously, to gallop." From the verb came the adverb randon "haphazardly, running off in one direction or another." The French word may be from Frankish rant "running," which would make it akin to English run. Another explanation connects random to an assumed German randa "edge" of a shield," perhaps because when a man was in battle he struck widely at enemies all around him with the edge of his shield. In statistics a sample of a population is said to be random if each member in the population has an equal chance of being chosen. To get such a sample you might have to "run around widely."

References

  1. R. B. Ash, Basic Probability Theory, Dover, 2008
  2. J. Bewersdorff, Luck, Logic, and White Lies: The Mathematics of Games, A K Peters, 2005
  3. G. J. Chaitin, The Limits of Mathematics, Springer, 1998
  4. D. Darling, The Universal Book of Mathematics, John Wiley & Sons, 2004
  5. B. S. Everitt, A. Skrondal, The Cambridge Dictionary of Statistics, Cambridge University Press, 2010 (Fourth edition)
  6. A. N. Kolmogorov, The Theory of Probability, in Mathematics: Its Content, Method and Meaning, Dover, 1999, II/229-264
  7. S. Schwartzman, The Words of Mathematics, MAA, 1994
  8. I. Stewart, Concepts of Modern Mathematics, Dover, 1995

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