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Bill Ricker

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

polymath

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.

cat herder

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.

Futbol Merkezi

Thank you for the information

Forumzamani

Thank you for the information

Many thanks

Good article. I will want a decent amount of time to think about your content!

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