Maths: How Binomial and Poisson Distributions Connect

Discrete probability distributions model situations where outcomes are counted rather than measured, and they sit at the heart of Number and Algebra in Maths AA SL. This topic brings together the binomial distribution, which counts successes in a fixed number of independent trials, and the Poisson distribution, which approximates rare events across many trials. Recognising when each applies — and how they connect through the mean λ = np — is the real skill being developed. The binomial formula P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ combines a counting term with the probability of a single outcome, so combinatorics and probability work together rather than separately. When n is large and p is small, the Poisson model with λ = np offers a simpler route, replacing factorials with e⁻λ λʳ / r!. Understanding these links helps you move fluently between exact and approximate models.