Learn9FM0Option · Further Statistics 1Poisson and binomial distributionsPoisson and normal approximations

Poisson and binomial distributions · Subtopic 4 of 4

Poisson and Normal Approximations

Two shortcuts. A binomial with many trials and a small success probability behaves like $\text{Po}(np)$ . A Poisson with a large mean behaves like $N(\lambda,\lambda)$ , as long as you remember the continuity correction.

Cambridge 9709 Paper 6Edexcel YMA01 WST02Edexcel 9FM0 OptionAQA 7367 Paper 3≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Approximate $B(n,p)$ by $\text{Po}(np)$ when $n>50$ and $np<5$ .
Approximate $\text{Po}(\lambda)$ by $N(\lambda,\lambda)$ when $\lambda>15$ .
Apply a continuity correction when using the normal approximation.
Choose the right approximation, and justify it.

Pause and think

$1.5\%$ of bolts are faulty. In a box of $200$ , $X\sim B(200,0.015)$ . Why is $P(X=4)=\binom{200}{4}(0.015)^4(0.985)^{196}$ awkward, and what could replace it?

Section 2 of 14

Core idea

Start hereShortcuts that work

$B(100,0.03)$ : computing $\binom{100}{2}0.03^2(0.97)^{98}$ is tedious. Rare events among many trials look Poisson. And a Poisson with a large mean looks like a bell curve. Each approximation saves time with very little error.

The big idea

$B(n,p)\approx\operatorname{Po}(np)$ for large $n$ , small $p$ ( $n>50$ , $np<5$ ). $\operatorname{Po}(\lambda)\approx N(\lambda,\lambda)$ for large $\lambda$ ( $\lambda>15$ ), with continuity correction.

1. Binomial → Poisson

B(100, 0.03) bars with Po(3) dots on top

np=3

. The Poisson dots sit almost exactly on the binomial bars.

Worked example3% faulty, batch of 100

P(exactly 2 faulty): exact and approximate.

Goal: Compare.
Strategy: $\lambda=np=3$ .

$B:\ \binom{100}2(0.03)^2(0.97)^{98}=0.2252$
Exact.
$\operatorname{Po}(3):\ e^{-3}\tfrac92=0.2240$
Approximate.

Answer

0.2252

0.2240

✓ CheckError under 1% ✓.

2. Poisson → Normal

Po(20) bars with the normal curve N(20, 20)

For a large mean the Poisson bars form a bell shape. Use a continuity correction:

P(X\le25)\approx P(Y<25.5)

Worked example $X\sim\operatorname{Po}(20)$ , $P(X\le25)$

Approximate.

Goal: Probability.
Strategy: $N(20,20)$ with correction.

$P(Y<25.5)$
Continuity correction.
$z=\tfrac{5.5}{\sqrt{20}}=1.230\Rightarrow0.891$
Standardise.

Answer

0.891

✓ CheckExact Poisson

0.888

✓.

Choosing an approximation

Drawing diagram…

Match the mean (and variance); correct for continuity whenever switching to the normal.

Pause and think

Which approximation for $B(200,0.01)$ ?

Summary

Rare events: Poisson.
Large mean: normal with correction.

Approximation: Simpler distribution giving nearly the same answers.

HistoryPoisson's own discovery. Poisson found his distribution in 1837 exactly as this lesson does: as the limit of the binomial when the number of trials grows and the probability shrinks, keeping

np

fixed. He was studying the number of wrongful convictions in French courts.

Fun factThree distributions, one curve. Binomial

\to

Poisson

\to

normal: a binomial with tiny

p

looks Poisson; a Poisson with large

\lambda

looks normal; and a binomial with large

n

and moderate

p

looks normal directly. They are all the same family of "counting random events" curves in different settings.

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