Probability Distributions (Binomial and Poisson)
This domain tests recognition of the binomial and Poisson situations, use of their probability formulae, and the mean and variance of each. Most questions are direct substitutions, so the marks turn on correct powers and correct parameters.
Binomial distribution
X ~ B(n, p) when there are n independent trials, each with two outcomes, and p is constant across trials, with q = 1 − p. Then P(X = r) = C(n, r)pʳqⁿ⁻ʳ for r = 0, 1, …, n. Mean = np, variance = npq, SD = √(npq). Since q < 1, the variance is always less than the mean.
Poisson distribution
X ~ P(m) when events occur randomly and independently with a constant average rate. P(X = r) = e⁻ᵐmʳ/r! for r = 0, 1, 2, …, with mean = variance = m and P(X = 0) = e⁻ᵐ. Poisson is the limiting form of the binomial when n is large, p is small and m = np.
Worked example
If X ~ B(4, 1/3), then P(X = 2) = C(4, 2)(1/3)²(2/3)² = 6 × (1/9) × (4/9) = 24/81 = 8/27. Mean = np = 4/3 and variance = npq = 4 × (1/3) × (2/3) = 8/9. Note that q is raised to n − r = 2, not to r.
Traps
- Raising q to the wrong power; it is n − r, not r.
- Omitting the binomial coefficient C(n, r).
- Using the given rate directly when the question asks about a different time interval; rescale m first.
- Applying Poisson when n is small and p is not small, where the approximation fails.