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About this book

We all know what randomness is. Or do we? Randomness turns out to be one of those concepts that works just fine on an everyday level, but becomes muddled upon close inspection. People familiar with quantum indeterminacy tell us that order is an illusion and that the world is fundamentally random. Yet these same people also say that randomness is an illusion: The appearance of randomness is only a sign of our ignorance and inability to detect the pattern.

By applying mathematical thinking, mathematician Edward Beltrami removes much of the vagueness that encumbers the concept of randomness. You will discover how to quantify what would otherwise remain elusive. As the book progresses, you will see how mathematics provides a framework for unifying how chance is interpreted across diverse disciplines. Communication engineering, computer science, philosophy, physics, and psychology join mathematics in the discourse to illuminate different facets of the same idea.

This book will provoke, entertain, and inform by challenging your ideas about randomness, providing different interpretations of what this concept means, and showing how order and randomness are really two sides of the same mysterious coin.

This second edition brings the question of randomness into the twenty-first century, adding compelling new topics such as quantum uncertainty, cognitive illusions caused by chance, Poisson processes, and Bayesian probability. An expanded technical notes section offers deeper explorations of a variety of mathematical concepts.

On the first edition:

I strongly recommend [What is Random?] to all who are interested in science and would like to see how the ideas of both theoretical mathematics and statistics have been observed and used in real life throughout history. The American Statistician

Table of Contents


1. The Taming of Chance

In the dim recesses of ancient history, the idea of chance was intertwined with that of fate. What was destined to be would be. Chance was personified, in the Roman Empire at least, by the Goddess Fortuna, who reigned as the sovereign of cynicism and fickleness. As Howard Patch puts it in his study of this Roman deity, “to men who felt that life shows no signs of fairness, and that what lies beyond is at best dubious, that the most you can do is take what comes your way, Fortuna represented a useful, if at times flippant, summary of the way things go.”
Edward Beltrami

2. Uncertainty and Information

A half-century ago Claude Shannon, mathematician and innovative engineer at what was then called the Bell Telephone Laboratories, formulated the idea of information content residing in a message, and, in a seminal paper of 1948, he established the discipline that became known as information theory. Though its influence is chiefly in communication engineering, information theory has come to play an important role in more recent years in elucidating the meaning of chance.
Edward Beltrami

3. Janus-Faced Randomness

The statistician M. Bartlett has introduced a simple step-by-step procedure for generating a random sequence that is so curious it compels us to examine it carefully since it will bring us to the very core of what makes randomness appear elusive.
Edward Beltrami

4. Algorithms, Information, and Chance

During the decade of the 1960s, several individuals independently arrived at a notion that a binary string is random if its shortest description is the string itself. Among the main protagonists in this story is the celebrated Russian mathematician Andrei Kolmogorov, whom we met earlier, and Gregory Chaitin, information theorist and computer scientist.
Edward Beltrami

5. The Edge of Randomness

Up to now I have been trying to capture the elusive notion of chance by looking at binary strings, the most ingenuous image of a succession of sensory events. Since most binary strings cannot be compressed, one would conclude that randomness is pervasive. However, the data streams of our consciousness do, in fact, exhibit some level of coherence. The brain processes sensory images and unravels the masses of data it receives, somehow anchoring our impressions by allowing patterns to emerge from the noise. If what we observe is not entirely random, it doesn’t mean, however, that it is deterministic. It is only that the correlations that appear in space and time lead to recognizable patterns that allow, as the poet Robert Frost puts it, “a temporary stay against confusion.” In the world that we observe, there is evidently a tension between order and disorder, between surprise and inevitability.
Edward Beltrami

6. Fooled by Chance

In the play Rosencrantz and Guildenstern by Tom Stoppard, a streak of 76 consecutive heads occur in a coin tossing game. The loser remarks that “A weaker man might be moved to re-examine his faith, if in nothing else than in the laws of probability”. And, later, “the equanimity of your average tosser of coins depends on a tendency which ensures he will not upset himself by losing too much nor upset his opponent by winning too often.”
Edward Beltrami


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