Poisson approximation to the binomialEdexcel International A Level Further Maths: Revision notes

For $X\sim B(n,p)$, finding $P(X=r)=\binom nr p^r(1-p)^{n-r}$ with large $n$ is awkward, and binomial tables do not cover large $n$. When $n$ is large and $p$ is small, the binomial distribution behaves very like a Poisson distribution, which is easier to use. Compare the means and variances. The binomial has $E(X)=np$ and $\text{Var}(X)=np(1-p)$. When $p$ is small, $1-p\approx1$, so the variance is close to the mean $np$. A Poisson distribution has its variance equal to its mean, so the two match well.

If $X\sim B(n,p)$ with $n$ large and $p$ small, then $X\approx\text{Po}(np).$ As a guide, use it when $n>50$ and $p<0.1$; the smaller $p$ and larger $n$, the better the fit. The parameter is $\lambda=np$, the binomial mean. Example: $X\sim B(80,0.05)$ gives $\lambda=80\times0.05=4$, so $X\approx\text{Po}(4)$.

Once you have $\text{Po}(np)$, use Poisson tables or the formula exactly as for any Poisson variable. Rewrite inequalities in terms of $P(X\le r)$. Worked example: $X\sim B(200,0.02)$. Then $X\approx\text{Po}(4)$, so $P(X\le2)=e^{-4}\left(1+4+\frac{4^2}{2}\right)=0.2381,$ $P(X>5)=1-P(X\le5)=1-0.7851=0.2149.$ For combined questions, such as 10 independent boxes, find the single-box probability with the approximation and then use the binomial or product rule for the boxes.

The approximation is a model, so the answer is not exact. Compare it with the exact binomial value if the question gives one. For $B(1000,0.002)$ the approximate value of $P(X\ge4)$ is $0.1429$ and the exact value is $0.1427$, which agree to 3 significant figures. The approximation gets worse when $p$ is larger, or $n$ is small. For example, $B(10,0.4)$ is not well modelled by a Poisson distribution, because the variance $np(1-p)=2.4$ is noticeably less than the mean $4$. A binomial variable can never exceed $n$, while a Poisson variable has no upper limit. This is harmless when $n$ is large and $p$ is small, as the chance of very large values is tiny.