Probability

De Moivre-Laplace

The normal approximation to a binomial, as n grows

De Moivre-Laplace

continuity correction

Key ideas

$\text{Binomial}(n, p) \approx N(np,\; np(1-p))$ for large n. This is the De Moivre-Laplace theorem.
Convergence is faster when $p$ is near 0.5 (symmetric) and slower near 0 or 1 (skewed).
Continuity correction improves the approximation by using $\mathbb{P}(k - 0.5 \leq X \leq k + 0.5)$ instead of the point PDF.
Total $|\text{error}|$ shrinks as $n$ grows, confirming convergence.