The binomial distribution
A binomial experiment has n independent trials, each with exactly two outcomes: success (probability p) or failure (probability q = 1 − p). If X is the number of successes, we write X ~ B(n, p).
Key formula
P(X = r) = ⁿCr pr (1 − p)n−r, r = 0, 1, …, n. Mean = np, variance = npq, standard deviation = √(npq).
Worked example
Worked example
X ~ B(5, 0.2). Find P(X = 2). P(X = 2) = ⁵C₂ (0.2)² (0.8)³ = 10 × 0.04 × 0.512 = 0.2048.
Mean, variance and standard deviation
For X ~ B(10, 0.3): mean = np = 10 × 0.3 = 3; variance = npq = 10 × 0.3 × 0.7 = 2.1; standard deviation = √2.1 ≈ 1.449.
Remember
- Trials must be independent with a fixed p.
- The powers of p and q always add to n.
- P(X ≥ 1) = 1 − P(X = 0) is often the quickest route.