11 Some paradoxes of randomness

In CHAPTER 10, we looked at some puzzles and paradoxes of probability arising from the subtle concepts of randomness and statistical independence. We want now to uncover some paradoxes of randomness.

We saw previously that statisticians are concerned with two aspects of the concept of randomness: defining the characteristics of a random sequence of numbers (as a theoretical underpinning of the concept of a ‘random variable’), and then defining operational methods for generating sequences of numbers that have these characteristics (as the basis for selecting a random sample, and for many other practical applications). The first of these aspects we can summarise by saying that a random sequence is, over a ‘very long’ run of numbers, notionally patternless and that, in a random sequence, each number is unpredictable from knowledge of those that came before.

We say ‘notionally patternless’, because there is no formal definition of a ‘very long’ run, and because there is no limit to the kinds of patterns we might want not to have in a random sequence. In practice, we have to limit ourselves to some small set of patterns to be excluded – specified so as to be appropriate to the practical context for which the random numbers are needed. We then look for a way of generating a sequence of numbers so that there is a low chance of such patterns turning up.

When it comes to unpredictability, we are similarly up against a practical constraint. In its ...

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