A friend of mine has been thinking about the well-known fact that many lists of seemingly random numbers (addresses, physical constants, populations) contain far disproportionately many initial 1's (compared to other initial digits, which decrease in frequency up to 9).

He noted that many physical constants are ratios—constants of proportionality, for example. So he calculated the probability that any two random numbers *x* and *y* between 0 and 1 (random with a flat distribution) have a ratio *y/x* that starts with a 1 (meaning its first non-zero digit: .00013 counts as an initial 1).

Cool answer: 1/3

The proof is left as an exercise (unless there's an outcry for it). Just a little tidbit to tide you over, since I haven't been posting much.

Unclear what relation this has to well known and very similar Benford's Law [1][2]. This distribution on 1..9 continues to differ besides 30.1% vs 1/3 (higher at both ends and lower in middle vs Benford's log d+1 - log d).

Demonstrating this exact equality to 1/3 is interesting in its own right, using either geometric proof, although monte carlo is unrigorously fun.

Thanks to your source for this for some lunchtime fun.

[1] http://en.wikipedia.org/wiki/Benford%27s_law

[2] http://mathworld.wolfram.com/BenfordsLaw.html

Posted by: Bill Ricker | October 16, 2007 at 01:02 PM

Thanks for those references; I didn't have time to hunt them down myself. I think the relation to Benford's law is this: It might at first seem implausible that even a list of scientific constants would obey the law. One might believe that certain lengths, charges, or times might have flatly distributed 1st digits (though I don't actually think they do). But even if some constants are flatly distributed, there are a large number of constants that are ratios (rates, constants of proportionality, etc.), and this fact shows that there's a reason to believe that the first digits of ratios are NOT flatly distributed. I think the point is that this result at least makes something like Benford's law more plausible.

Posted by: polymath | October 17, 2007 at 07:17 PM

You can do this game for the first k decimals and ask about the probability that the coefficient of the scientific notation of y/x starts with a given sequence of k digits. The result is that the highest probability will be for scientific notations with a coefficient that is 1.000000... . On the other hand, the expected value of the coeffficient will be approx 4.02921.

For the exponent of the scientific notation of y/x, the most probable exponent is either 0 or -1, and the expected value of the exponent is -0.5.

I would be interested to see the results for other expressions such as xy or x+y, etc.

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