I actually think I know this one without having to look it up! Most people think that the digits in numbers are uniformly distributed but that isn't true. The first digit of numbers follow what is called Benford's Law. Benford's Law shows that 1 is the most likely first digit and 9 the least likely first digit (the probability monotonically decreases from 1 to 9).
I have seen the probabilities worked out for second digits and the relationship is much weaker even though 1 is the most likely second digit. I would guess that by the time you get to the third digit that the probabilities are uniform.
Data sampled randomly (or pseudo-randomly) in this range should follow Benford's law. However, when we include every digit in the range, then we see an almost-uniformity of all nonzero digits.
I notice that if the range had been "1 to 999, inclusive" then all digits except 0 would have 300 occurrences.
Hi BS King,
ReplyDeleteI actually think I know this one without having to look it up! Most people think that the digits in numbers are uniformly distributed but that isn't true. The first digit of numbers follow what is called Benford's Law. Benford's Law shows that 1 is the most likely first digit and 9 the least likely first digit (the probability monotonically decreases from 1 to 9).
I have seen the probabilities worked out for second digits and the relationship is much weaker even though 1 is the most likely second digit. I would guess that by the time you get to the third digit that the probabilities are uniform.
I hope that makes sense.
Glenn
The most-frequent digit in that range is 1.
ReplyDelete(I was able to write a simple computer program that evaluated the question for an arbitrary range...is that considered cheating?)
Details:
Delete0 (192 times)
1 (301 times)
2 (300 times)
3 (300 times)
4 (300 times)
5 (300 times)
6 (300 times)
7 (300 times)
8 (300 times)
9 (300 times)
Data sampled randomly (or pseudo-randomly) in this range should follow Benford's law. However, when we include every digit in the range, then we see an almost-uniformity of all nonzero digits.
I notice that if the range had been "1 to 999, inclusive" then all digits except 0 would have 300 occurrences.
Reminded me of this: http://xkcd.com/1103/
ReplyDeleteIn a belated first attempt at GeoGuesser, I only got 8410 points.
ReplyDelete