[chongli]

That's only the most limited version of the theorem which has since been renamed the de Moivre-Laplace theorem [1]. The rabbit hole goes much deeper when you talk about the most general form which works for any set of independent and identically distributed random variables, not just binomial random variables.

[1] https://en.wikipedia.org/wiki/De_Moivre–Laplace_theorem

[adrianN]

Sure, but do you need the most general form to build some intuition about p-values? I don't think so.

[chongli]

That’s moving the goalposts. The original claim was about getting a complete picture. A full understanding of the “why” of statistics going all the way to the bottom.