Quote Originally Posted by DJAbacus View Post
Yes, but we are not taking the amount of hands into consideration here, only the amount of trials.

So if we did 100,000 trials of 50K hands

and then 100,000 trials of 250K hands

and then 100,000 trials of 1M hands

The shape of these 3 normal distributions would look very different.
This is a different set of distributions entirely.

Now you're talking about the average of the averages... or the mean of the means.
E.g. You calculate the mean from one sample of 50k hands.
Then you do that 100,000 times.

Now you want to know the distribution of the means. And you want to know the mean of that.
You are asking about the mean of means.
(just to clarify, 'cause it's an odd thing to understand the first time through.)

The statistics which apply to statistics is a different set of math again. We're basically invoking the law of large numbers and regression to the mean.

Which yields a convergent distribution, but not necessarily convergent with 0 variance - in this case, it would be convergent with var(X) = sqrt(n)/2.
So here's a new topic we haven't touched on yet. Sometimes perfect information yields a probability distribution. Meaning perfect information is still embedded with uncertainty.
In this case, it's because we're getting meta about our stats, and we're being very clever and careful about quantifying what we "know" and what we "expect."