The Poisson Approximation of the Binomial Distribution

When the number of trials is large and the success probability is small, the binomial distribution can be approximated by a Poisson distribution with parameter $\lambda = np$ . This lesson develops the approximation formula, applies it to concrete word problems, and extends it to cumulative probabilities such as $P(X \geq k)$ and $P(X \leq k)$ .

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Approximating the Binomial With a Poisson

The binomial pmf $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$ is unwieldy to compute by hand when $n$ is large. When $p$ is also small, the Poisson approximation provides a clean alternative.

Poisson Approximation to the Binomial. If $X \sim \mathrm{Binomial}(n,p)$ with $n$ large and $p$ small, then

P(X = k) \;\approx\; \dfrac{e^{-\lambda}\, \lambda^k}{k!}, \qquad \text{where } \lambda = np.

Notice that both distributions share the same mean ( $np = \lambda$ ). As $n \to \infty$ and $p \to 0$ with $\lambda = np$ held fixed, the binomial pmf converges to the Poisson pmf. A common rule of thumb is $n \geq 20$ and $p \leq 0.05$ .

Illustration. Let $X \sim \mathrm{Binomial}(100, 0.03)$ . Setting $\lambda = np = 3{:}$

P(X = 1) \;\approx\; \dfrac{e^{-3} \cdot 3^1}{1!} = 3e^{-3} \approx 0.149.

The exact binomial probability is $0.147,$ so the approximation is accurate to three decimal places.