Quote Originally Posted by Zangief
Quote Originally Posted by Pyroxene
But by the time I try it 100 times it is unlikely that I will be far from 40% AND (this is important) by the time I try it 1000 times I WILL be closer to 40% than I was at 100 tries AND by the time I try it 10,000 times I WILL be closer still.

...

Technical Note: The I WILL part actually comes from a thing called The Strong Law of Large Numbers, but it is very similar to The Law of Large Numbers.
I don't think I believe this. I think eventually you should get closer to 40% than where you start from, but I don't believe that you can say you will be closer at 1000 tries than at 100. You always wiggle around the exact percentage.

What if, on this particular occasion, you hit 40% exactly on 100 tries (40/100). I think chances are good that you could waiver off of that 40%, even by just a little, by the time you get to 1000 tries (say, 399/1000).

This doesn't follow that the law that YOU WILL always be closer when going from 100 to 1000 or 1000 to 10,000.

I know I haven't proved anything, but I believe what I am describing can and does happen.
I cannot express the equation of the Strong Law of Large Numbers as it involves a limit of a sumation from 1 to infinity and I do not know how to draw the symbols using the editor.

But to put it in non-math terms, the "Law of Large Numbers" says that as we take more and more independent samples we tend to approach the expected distribution of those sample. Thus, it is unlikely that after significantly more samples the actually distribution would be further from the expected distribution.

The "Strong Law of Large Numbers" goes a bit further. While the "Law of Large Numbers" says that moving further from the expected distribution is unlikely, the "Strong Law of Large Numbers" says it does not happen. To put it in slightly more mathematic terms, it says that if the difference between your expected distribution and your actual distribution is epsilon after N samples, there exists some number of samples M, with M > N, where the difference between your actual and expected distribution is not only less than or equal to epsilon, but also that beyond M samples the difference will NEVER be greater than epsilon again. And that is where the certainty part of the statement comes from. I am making an assumption that in our flush draw sample that increasing the samples by an order of magnitude is sufficient to achieve the effect that the "Strong Law of Large Numbers" describes. I am open to any arguments that an order of magnitude is insufficient, as I cannot prove it. However, the "Strong Law of Large Numbers" dictates that there is such a number, and there is a number beyond that where you will be forever closer still, and so on and so on.

And just to explicitly state the obvious end of this, the "Strong Law of Large Numbers" says that the probability of the measured distribution being equal to the expected distribution in an infinite number of samples is 1. Thus, in an infinite number of samples, you make that 37% flush draw exactly 37% of the time; without exception.

Editted: Confused when epsilon would be less and when it would be less than or equal to. And fixed a glaring spelling error, because speiling is haurd.