# Is the average of many “random” numbers useful information?

Ok, so I found this site: http://tweetcracker.com/. Essentially, people just tweet 10 digit numbers in hopes it is the correct number (like lottery, except free).

I heard that if you took all the numbers in the world (e.g. from bank statements, newspapers, stocks, sports scores, etc...) that you would get a logarithmic distribution, with "1" being on top (p=30.1%) and "9" on bottom, consecutively decreasing logarithmically. (Benford's Law)

I also heard that if you averaged all the guesses from a game of "guess the number of gum balls to win a prize" that you get a bell curve distribution and the correct answer is usually very close to the mean (average). This at least uses spatial skills, while Benford's Law and my tweetcracker situation are essentially totally random.

I thought about taking all the guesses and averaging them, but I suppose I would get some reflection of Benford's Law? I might just write a java program to process the data for me (just for the hell of it). I was just wondering what more experienced mathematicians thought about this? Any possible ways of narrowing down solution besides brute force? Those were all the ideas I could think of, intriguing, though randomness is a pretty fundamental principle so I guess if someone somehow "broke" it, then lotteries would not be around!! :) Nevertheless interesting, haha.

Averaging won't get you closer to a completely unseen random number. With a gumball machine, people can actually see the number even if they can't count it, so it makes sense that guesses would group around the real value. A random number, on the other hand, is hidden in the dark and no one knows if it's closer to $0$ or $10^{10}-1$ or anywhere in between, so the guesses won't group around anywhere in particular. If the guesses are computer-generated they are simply going to exemplify a boring uniform distribution over 10-digit integers.
If you partition the numbers between $0$ and $10^{10}-1$ into those with first digit $1, 2, \cdots, 9$, it's not hard to see each of the cells (parts of the partition) will be in one-to-one correspondance with each other (simply alter the first digit of every number in one part to map it to another part of the partition). There's no reason to suspect any guess is more likely than any other, so first digits are equidistributed and won't exemplify Benford's Law. (This law applies in the real world only when statistics are pulled from a number of distributions individually spread over different orders of magnitude.)