Poisson Approximation to the Binomial Distribution — Mathematics

Introduction

In many real-world scenarios, a binomial distribution $\text{B}(n, p)$ arises where $n$ is very large and $p$ is very small — for example, the number of defective components in a large batch, or the number of rare genetic mutations in a population. In such cases, computing exact binomial probabilities becomes computationally burdensome. The Poisson distribution provides a convenient and accurate approximation, and the 9709 exam regularly tests whether candidates can recognise when this approximation is valid, state the approximating distribution correctly, and use it to calculate probabilities.

Core Concept

Recall that if $X \sim \text{B}(n, p)$ , then:

P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}

When $n$ is large and $p$ is small, it can be shown algebraically that this expression converges towards the Poisson probability formula with $\lambda = np$ . Intuitively, the mean of the binomial is $np$ and its variance is $np(1-p)$ . When $p$ is very small, $(1-p) \approx 1$ , so the variance $np(1-p) \approx np = \lambda$ . This matches the Poisson distribution, where the mean and variance are both equal to $\lambda$ — which is precisely why the approximation works.

Conditions for the approximation to be valid:

Condition	Guideline
$n$ is large	$n > 50$
$p$ is small	$np < 5$ (approximately)

When both conditions are met, we approximate:

X \sim \text{B}(n, p) \approx Y \sim \text{Po}(\lambda), \quad \text{where } \lambda = np

Important: You must always state the approximating distribution and the value of $\lambda$ explicitly in your working — examiners award method marks for this.

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Introduction

Core Concept

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