A binomial variable counts successes in n independent trials, each with success probability p. Set n and p, then read off P(X = k), the mean μ = np, and the SD σ = √(np(1−p)). Watch the shape skew as p leaves 0.5.
■ PMF bar ■ selected outcome(s) — normal approx.
Try: set p = 0.10, n = 10 — the PMF is right-skewed and the normal overlay fits badly (np = 1 < 10). Now push n = 40, p = 0.50: the bars go symmetric and the bell curve traces them, because np = 20 and n(1−p) = 20 both clear 10.