trials = hands

Even if not... all you did was say, OK, I want to generate thousands of distributions, then overlap them all.

That result, taking the trials in separate sets of experiments, will have higher variance than if you took all of the trials as a single experiment.

And none of them would yield a normal distribution.
The normal distribution is for continuous statistics, and this is discreet statistics.

The distribution here is the Binomial distribution, for which I linked to the wiki above.

***
You're seriously digging in your heels, when you should be taking notes.