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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn7367Paper 3 · Statistics optionPoisson distributionPoisson probabilities

Poisson distribution · Subtopic 1 of 3

Poisson Probabilities

Rare, random events in a fixed interval, such as calls per hour, typos per page or meteors per night, follow the Poisson distribution. One number, the average rate $\lambda$ , sets everything: $P(X=r)=e^{-\lambda}\dfrac{\lambda^r}{r!}$ .

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

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Section 1 of 14

Learning objectives

State the conditions for a Poisson model in context.
Calculate $P(X=r)=e^{-\lambda}\dfrac{\lambda^r}{r!}$ and cumulative probabilities.
Know that the mean and variance of $\text{Po}(\lambda)$ are both $\lambda$ .
Use "mean $\approx$ variance" in data as evidence for a Poisson model.

Pause and think

A help desk gets on average $3$ calls an hour, at random. Is it possible to get $10$ calls in one hour? Is there a maximum number of calls?

Section 2 of 14

Core idea

Start hereCounting random events

A call centre averages 3 calls a minute. Some minutes get none, some get 7. Calls arrive randomly, one at a time, independently. What's the chance of exactly 5 in a minute? Any situation like this, counting random events in a fixed interval, follows the Poisson distribution.

The big idea

$X\sim\operatorname{Po}(\lambda)$ counts events occurring singly, independently and at a constant average rate $\lambda$ per interval. $P(X=r)=e^{-\lambda}\dfrac{\lambda^r}{r!}$ ; mean = variance = $\lambda$ .

1. Conditions

Events occur singly (not two at once).
Independently.
At a constant average rate.

2. The formula

Po(λ): the shape as the average rate changes

λ = 1.0

Small

\lambda

: most likely to see 0 or 1 events. As

\lambda

grows the peak moves right and the shape becomes more symmetric. The mean and the variance are both

\lambda

P(X=r)=e^{-\lambda}\frac{\lambda^r}{r!},\quad r=0,1,2,\ldots

$\lambda$: mean number of events per interval.
$e$: $2.718\ldots$

The probabilities add to 1

$\sum_{r=0}^\infty e^{-\lambda}\frac{\lambda^r}{r!}=e^{-\lambda}\left(1+\lambda+\frac{\lambda^2}{2!}+\cdots\right)$
Factor out $e^{-\lambda}$ .
$=e^{-\lambda}e^{\lambda}=1$
The bracket is the series for $e^\lambda$ .

(Poisson arises as the limit of

B(n,p)

with

n\to\infty

np=\lambda

: many tiny chances of an event.)

Worked example $X\sim\operatorname{Po}(3)$

Find $P(X=2)$ , $P(X\le2)$ , $P(X\ge3)$ .

Goal: Three probabilities.
Strategy: Formula; complement.

$P(X=2)=e^{-3}\tfrac92=0.224$
$r=2$ .
$P(X\le2)=e^{-3}(1+3+4.5)=0.423$
$r=0,1,2$ .
$P(X\ge3)=1-0.423=0.577$
Complement.

Answer

0.224

0.423

0.577

✓ CheckEach term from the last:

P(r)=P(r-1)\times\frac\lambda r

0.0498\to0.149\to0.224

✓.

TrickIs Poisson plausible? If sample mean ≈ sample variance, a Poisson model is reasonable.

Pause and think

$X\sim\operatorname{Po}(1.5)$ : $P(X=0)$ ?

Summary

Single, independent, constant rate.
$e^{-\lambda}\lambda^r/r!$ ; mean = variance.

Poisson distribution: Model for counts of random events.

Poisson probabilities for X ~ Po(3)

P(X=x)=e^{-3}\dfrac{3^x}{x!}

. The distribution peaks near the mean,

3

, with a tail to the right.

HistoryDeaths by horse kick. Siméon Denis Poisson published the distribution in 1837, but it became famous in 1898 when Ladislaus Bortkiewicz showed it fitted the number of Prussian cavalry soldiers killed by horse kicks each year, over

20

years and

14

army corps. Rare, random, independent events: a perfect Poisson process.

Fun factThe "fish" distribution. "Poisson" is French for "fish", so students sometimes call it the fish distribution. Fittingly, ecologists use it to model the number of fish caught per net, or plants per square of meadow, to test whether things are scattered at random or clumped together.

Section 3 of 14

Key formulae

Poisson probability

P(X=r)=e^{-\lambda}\frac{\lambda^r}{r!},\qquad r=0,1,2,\ldots

Mean and variance

E(X)=\text{Var}(X)=\lambda

Cumulative

P(X\le2)=e^{-\lambda}\left(1+\lambda+\frac{\lambda^2}{2!}\right)

TrickEach term from the last.

P(X=r)=P(X=r-1)\times\dfrac{\lambda}{r}

. Start from

P(X=0)=e^{-\lambda}

and multiply by

\tfrac\lambda1,\tfrac\lambda2,\tfrac\lambda3,\ldots

to get the next terms: no factorials or big powers needed, and it is easy on a calculator.

Section 4 of 14

Worked examples

Example 1 · Less than0 / 2 steps

X\sim\text{Po}(4.5)

. Find

P(X<3)

Example 2 · A range0 / 2 steps

X\sim\text{Po}(3)

. Find

P(2\le X\le4)

Example 3 · Checking a model0 / 2 steps

The numbers of emails arriving per minute have sample mean

3.2

and variance

3.1

. Comment on a Poisson model.

Section 5 of 14

Explore the graph

The shape of Po(λ)

Loading the interactive graph…

Try this

The curve passes through $P(X=r)$ for $X\sim\text{Po}(l)$ at each whole $r$ ; the vertical line is the mean $l$ .
For small $l$ the distribution is very skewed. What happens to its shape as $l$ grows?
For whole-number $l$ , check that $P(X=l-1)=P(X=l)$ : the peak is shared by two values.

As $\lambda$ increases the Poisson becomes bell-shaped, which is why a normal approximation works for large $\lambda$ .

In the real worldPlanning call centres. If calls arrive as

\text{Po}(12)

per

5

minutes, managers can calculate the chance of more than

20

calls in a

5

-minute slot and schedule enough staff to keep waiting times down. The same reasoning sizes hospital emergency departments and supermarket checkouts.

Section 6 of 14

Common mistakes

Incorrect

P(X<3)

includes

P(3)

Correct

P(0)+P(1)+P(2)

"Less than

3

" stops at

2

IncorrectForgetting

P(X=0)

in a cumulative sum

CorrectStart at

r=0

Zero events is a possible outcome.

IncorrectStandard deviation

=\lambda

Correct

\sqrt\lambda

The variance is

\lambda

IncorrectConditions stated without context

Correct"Calls arrive independently of each other…"

Marks require the context of the question.

Binomial

I need $n$ and $p$

Poisson

just give me $\lambda$

Poisson

that's it. That's the whole personality.

Section 7 of 14

Exam tip

Exam tipConditions in context. When asked for conditions, write them about the situation: "faults occur at random in the cloth, independently of each other, at a constant average rate per metre". Generic statements without the context usually earn nothing.

Section 8 of 14

Guided practice

SA-9709-P6-PO-G012 marksL1 FoundationCalculatorProbabilities

X\sim\text{Po}(3)

. Find

P(X=2)

, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-9709-P6-PO1-0011 markL1 FoundationCalculatorProbabilities

X\sim\text{Po}(2)

. Find

P(X=0)

, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-PO-0012 marksL1 FoundationCalculatorProbabilities

X\sim\text{Po}(3)

. Find

P(X\le2)

, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-PO1-0022 marksL2 StandardCalculatorProbabilities

X\sim\text{Po}(4.5)

. Find

P(X<3)

, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-PO-0023 marksL2 StandardCalculatorProbabilities

X\sim\text{Po}(2.5)

. Find

P(X\ge3)

, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-PO1-0032 marksL2 StandardCalculatorProbabilities

X\sim\text{Po}(1.8)

. Find

P(X>1)

, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P6-PO-0082 marksL1 FoundationCalculatorProbabilities

A Poisson variable has variance

5

. Find

P(X=0)

, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

More questions on every skill in this subtopic are in the question bank under Practise.

Section 10 of 14

Challenge

SA-9709-P6-PO1-0063 marksL3 ExamCalculatorProbabilities

X\sim\text{Po}(3)

. Find

P(2\le X\le4)

, correct to 3 significant figures.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

Poisson: random, independent, single events at a constant average rate.
$P(X=r)=e^{-\lambda}\dfrac{\lambda^r}{r!}$ .
Mean $=$ variance $=\lambda$ .
Cumulative: add from $r=0$ ; "at least": use the complement.

Section 12 of 14

Quick quiz

Q1 · SA-9709-P6-PO-Q011 markL1 FoundationCalculatorProbabilities

For

X\sim\text{Po}(\lambda)

\text{Var}(X)

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-9709-P6-PO-0091 markL1 FoundationCalculatorConditions

A Poisson model is suitable when events occur

Salam Academy Original Practice · Proposed mark scheme

Q3 · SA-9709-P6-PO1-0041 markL1 FoundationCalculatorConditions

Which of these is most likely to be well modelled by a Poisson distribution?

Salam Academy Original Practice · Proposed mark scheme

Q4 · SA-9709-P6-PO1-0051 markL1 FoundationCalculatorConditions

Counts of cars passing a point per minute have mean

3.2

and variance

8.9

. What does this suggest about a Poisson model?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 7367 A-level Further Mathematics
Where: Further Statistics · Discrete distributions
Expected: The Poisson distribution, its conditions, mean and variance, and sums of independent Poisson variables.
Calculator: Permitted on all papers; a calculator with statistical distribution functions is expected.
Typical style: Further Statistics option; calculators with distribution functions are expected.
Open in: 7367 · Paper 3

Section 14 of 14

Past paper questions

5 questions (38 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

9709/62 · Oct/Nov 2025Question 18 marks
Conditions for a Poisson model, and probabilities over 4 and 20 weeks with an approximation.
Question paper, page 2 Mark scheme
9709/62 · May/June 2025Question 34 marks
Poisson probability in terms of e, and the value of n making two probabilities equal.
Question paper, page 5 Mark scheme
9709/62 · Oct/Nov 2024Question 511 marks
Poisson probabilities, sums and a normal approximation.
Question paper, page 6 Mark scheme
9709/62 · May/June 2023Question 25 marks
Poisson mean and variance, when Poisson approximates binomial, and a tail probability.
Question paper, page 3 Mark scheme
9709/62 · May/June 2023Question 410 marks
Poisson conditions, probabilities over different periods, and a Poisson total.
Question paper, page 6 Mark scheme

All 9709 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 7367, First teaching 2017 (official specification).