Saturday, July 20, 2002

Joe writes:

I read recently that in any list of random numbers, the first number (and the second and third, to a lesser extent) are not randomly distributed, that higher digits are more likely (that the probability that the first digit will be n or less is not n/9 but rather log n+1.) This makes no sense to me. Can you put it in layman's terms?

Random numbers are, by definition, randomly distributed. But there are many possible distributions. With infinitely many numbers, it is not enough to say that all numbers are equally likely. One distribution is the uniform distribution of the real numbers from 0 to 1. This has numbers in [0,.1], [.1,.2], ..., each having probability 1/10. Another distribution is the bell curve.

Looking at leading digits of numbers found in nature leads to a distribution with a multiplicative invariance, not an additive invariance. Simon Newcomb noticed that the pages on his log tables were more worn at the beginning than at the end, and he published a logarithmic distribution in 1881. It looks like numbers on a slide rule. For more explanation, look for the First Digit Law.

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