from typing import Tuple
import math as m
def normal_approximation_to_binomial(n: int, p: float) -> Tuple[float, float]:
    mu = p * n
    sigma = m.sqrt(p * (1 - p) * n)

    return mu, sigma

The change of standard deviation with the incrase of number of trials.

for i in range(10):
    print(i, normal_approximation_to_binomial(10**i, 0.5))
0 (0.5, 0.5)
1 (5.0, 1.5811388300841898)
2 (50.0, 5.0)
3 (500.0, 15.811388300841896)
4 (5000.0, 50.0)
5 (50000.0, 158.11388300841898)
6 (500000.0, 500.0)
7 (5000000.0, 1581.1388300841897)
8 (50000000.0, 5000.0)
9 (500000000.0, 15811.388300841896)