The binomial distribution · Grade 12
In Grade 12 you learn: binomial probabilities.
The binomial distribution counts the number of successes in a fixed number of independent trials. Each trial has the same probability of success. If X is the number of successes in n trials with probability p, then X ~ B(n, p).
Key facts
- Conditions: a fixed number of trials n, two outcomes each time, a constant probability p, and independent trials.
- P(X = r) = nCr × p^r × (1 − p)^(n − r).
- P(X ≤ r): add P(X = 0) up to P(X = r). Or use cumulative tables or a calculator.
- P(X ≥ r) = 1 − P(X ≤ r − 1).
- Mean: E(X) = np. Variance: Var(X) = np(1 − p).
- P(at least one success) = 1 − (1 − p)^n.