The law of large numbers applies here.

We're sampling the payout table's probability distribution many times, keeping a cumulative running total.

The set of the cumulative running totals' values at the end of the trials, no matter how long the trials, will create a distribution of it's own. Do you see?

So the law of large numbers says that if we run trials of any length, what we measure of any statistic... those measurements will create a normal distribution the more and more of them we take.