Edexcel A-level Further Maths revision notes

Poisson and binomial distributions

Section FS1-2

Year 1

3 specification points

Notes and three levels of exam-style practice for each registered specification point in this section.

Checked against Edexcel 9FM0 section FS1-2

Checked against Edexcel 9FM0 section FS1-2. Review basis: the qualification registry sourced from the Pearson Edexcel Level 3 Advanced GCE in Further Mathematics (9FM0) specification; registry verification recorded 17 July 2026.

How this checking works

In the exam: Formulae booklet provided · calculator allowed in every paper

Open the printable pack

FS1-2.1

The Poisson distribution. The additive property of Poisson distributions.

Notes

Evidence from your answers: none yet

Your confidence:

A self-report of how sure you feel. It does not measure mastery. Evidence from your answers reaches secure after the latest Tier 2/3 attempt is correct, with three correct distinct drills across at least two dates and two practice sources.

Explanation

A Poisson distribution models a count of events occurring independently at a constant mean rate: $P(X=x)=e^{-\lambda}\dfrac{\lambda^x}{x!}$ for $x=0,1,2,\ldots$ . Scale $\lambda$ in proportion to the length or size of the observation interval.
If $X\sim\operatorname{Po}(\lambda)$ and $Y\sim\operatorname{Po}(\mu)$ are independent, then $X+Y\sim\operatorname{Po}(\lambda+\mu)$ .
This additive property also combines disjoint intervals of a Poisson process.
A real-world answer may require critical comment on constant rate and independence.
Examiners expect the parameter to match the requested interval and cumulative probabilities to use the correct complement and inclusive boundary.

A typical discrete, right-skewed Poisson probability mass function.

Worked example

Calls arrive at mean rate $1.4$ per $10$ minutes. Find the probability of at least $5$ calls in $30$ minutes.

1.For $30$ minutes, $\lambda=3(1.4)=4.2$ .
2.Let $X\sim\operatorname{Po}(4.2)$ .
3. $P(X\geq5)=1-P(X\leq4)=0.4102$ to $4$ significant figures.

Answer: The probability is $0.4102$ .

Common mistakes

Don't use the rate for $10$ minutes as the parameter for a $30$ -minute count.
Don't add parameters for counts that are not independent.
Don't calculate $1-P(X\leq5)$ for the event $X\geq5$ .

Exam tip

Write the scaled distribution before using a calculator; this makes the interval conversion explicit.

Tier 1 · Easy

ORIGINAL

A sensor records a Poisson-distributed number of amber flashes with mean

2.7

per minute. Find the probability of no amber flashes in one minute.

(2)

(Total for Question 1 is 2 marks)

Tier 2 · Standard

ORIGINAL

Independent counts

A

and

B

have distributions

\operatorname{Po}(3.2)

and

\operatorname{Po}(1.7)

. Find

P(A+B\geq7)

(3)

(Total for Question 1 is 3 marks)

Tier 3 · Hard

ORIGINAL

Inspection pings are modelled by a Poisson process at a mean rate of

0.8

per

10

minutes. Find the probability of exactly

2

pings in the first

15

minutes and at most

2

pings in the next

30

minutes. State two assumptions needed for the model and one feature of the real process that would make it unsuitable.

(8)

(Total for Question 1 is 8 marks)

Your progress and exam materials

This section: Evidence from your answers: 0/3 secureYour confidence: 0 self-rated secureTracker status: 0/3 secure, 0 shaky, 3 unseen

Overall: Evidence from your answers: 0/116 secureYour confidence: 0 self-rated secureTracker status: 0/116 secure, 0 shaky, 116 unseen

Progress is saved on this device for guests and accounts right now; cross-device account sync is not live yet.

FS1-2.2

The mean and variance of the binomial distribution and the Poisson distribution.

Notes

Evidence from your answers: none yet

Your confidence:

Explanation

For $X\sim\operatorname{Bin}(n,p)$ , $E(X)=np$ and $\operatorname{Var}(X)=np(1-p)$ . For $Y\sim\operatorname{Po}(\lambda)$ , both mean and variance equal $\lambda$ .
Under a linear transformation, $E(aX+b)=aE(X)+b$ but $\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X)$ because adding a constant changes location, not spread.
These relationships can recover unknown distribution parameters from given means or variances and help compare models with observed data.
The binomial parameter $n$ must be a positive integer and $0\leq p\leq1$ .
Examiners expect simultaneous parameter equations to be solved exactly where possible and a standard deviation to be the positive square root of variance.

Worked example

A binomial random variable has mean $24$ and variance $18$ . Find $n$ and $p$ .

1. $np=24$ and $np(1-p)=18$ .
2.Divide the variance equation by the mean equation: $1-p=\dfrac{18}{24}=\dfrac34$ .
3.Thus $p=\dfrac14$ and $n=24/(1/4)=96$ .

Answer: $n=96$ and $p=\dfrac14$ .

Common mistakes

Don't use $np$ as both the mean and variance of a binomial distribution.
Don't add $b$ to the variance when transforming $aX+b$ .
Don't multiply variance by $a$ rather than $a^2$ .

Exam tip

When both binomial mean and variance are given, divide the equations first to isolate $1-p$ .

Tier 1 · Easy

ORIGINAL

Given

X\sim\operatorname{Bin}(80,0.15)

, find

E(X)

and

\operatorname{Var}(X)

(2)

(Total for Question 1 is 2 marks)

Tier 2 · Standard

ORIGINAL

The random variable

Y

is Poisson and

\operatorname{Var}(2Y+3)=28

. Find the parameter of

Y

and

E(2Y+3)

(4)

(Total for Question 1 is 4 marks)

Tier 3 · Hard

ORIGINAL

A binomial random variable

X

has mean

18

and variance

13.5

. Determine

n

and

p

. Hence find

P(X=18)

, giving your answer to

4

significant figures.

(6)

(Total for Question 1 is 6 marks)

FS1-2.3

The use of the Poisson distribution as an approximation to the binomial distribution.

Notes

Evidence from your answers: none yet

Your confidence:

Explanation

When $n$ is large and $p$ is small, $X\sim\operatorname{Bin}(n,p)$ may be approximated by $Y\sim\operatorname{Po}(np)$ . State why the conditions are reasonable and set $\lambda=np$ .
The event boundary remains discrete and unchanged: no continuity correction is used for a binomial-to-Poisson approximation.
Calculator cumulative functions or complements can then evaluate the probability.
The approximation becomes less reliable when $p$ is not small or when the binomial trials are not independent with constant success probability.
Examiners expect approximation notation or clear wording, the value of $\lambda$ , and a probability answer labelled approximate rather than exact.

Worked example

Let $X\sim\operatorname{Bin}(800,0.006)$ . Use a Poisson approximation to estimate $P(X\geq7)$ .

1. $n$ is large, $p$ is small and $np=4.8$ , so use $Y\sim\operatorname{Po}(4.8)$ .
2. $P(X\geq7)\approx P(Y\geq7)$ .
3. $P(Y\geq7)=1-P(Y\leq6)=0.2092$ to $4$ significant figures.

Answer: $P(X\geq7)\approx0.2092$ .

Common mistakes

Don't use $\lambda=p$ instead of $\lambda=np$ .
Don't apply a continuity correction even though both distributions are discrete.
Don't quote the approximation without checking that $n$ is large and $p$ is small.

Exam tip

A full approximation line should state the conditions, $Y\sim\operatorname{Po}(np)$ and the unchanged event.

Tier 1 · Easy

ORIGINAL

Let

X\sim\operatorname{Bin}(900,0.004)

. State a suitable Poisson approximation and use it to estimate

P(X=0)

(3)

(Total for Question 1 is 3 marks)

Tier 2 · Standard

ORIGINAL

The number of flawed seals in a batch has distribution

X\sim\operatorname{Bin}(400,0.01)

. Use a Poisson approximation to estimate

P(X\geq6)

(4)

(Total for Question 1 is 4 marks)

Tier 3 · Hard

ORIGINAL

For

X\sim\operatorname{Bin}(200,0.015)

, calculate

P(X\leq2)

exactly and by a Poisson approximation. Find the percentage error of the approximation relative to the exact value.

(6)

(Total for Question 1 is 6 marks)

Want help turning these notes into marks?

Bring a tricky specification point or a recent answer, and we can work through the method and exam wording together.

Book a free intro call