[TimonKnigge]

Your 'approximation' is doing a lot of the work here.. Since you have a binomial distribution why not just use it directly?

(1 - 2e-6)^(3e6) ≈ 0.002

So about 0.2%. Still highly unlikely but orders of magnitude more likely than what your normal distribution-detour gave.

[petrogradphilos]

> (1 - 2e-6)^(3e6) ≈ 0.002

Using the binomial directly is a good way to get the probability of 0 heads. Note, though, that the U.S. population is in the neighborhood of 300 million, not 3 million (as you seem to have used).

(1 - 2⋅10⁻⁶)^(3⋅10⁸) ≈ 10⁻²⁶¹

https://www.wolframalpha.com/input/?i=%281+-+2+*+10%5E-6%29%...