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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 optionDiscrete random variablesProbability distributions

Discrete random variables · Subtopic 1 of 3

Probability Distributions

A random variable turns the outcome of an experiment into a number. Listing each possible value with its probability gives a distribution table, and the probabilities must add up to $1$ .

Cambridge 9709 Paper 5Edexcel 9MA0 Paper 3Edexcel 9FM0 OptionAQA 7357 Paper 3AQA 7367 Paper 3≈ 30 minRevision 5 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Understand a discrete random variable as a numerical outcome with a set of probabilities.
Draw up a probability distribution table from a situation.
Use $\Sigma P(X=x)=1$ to find unknown probabilities or constants.
Find probabilities such as $P(X\ge2)$ from a table.

Pause and think

Two coins are tossed and $X$ is the number of heads. What values can $X$ take, and are they equally likely?

Section 2 of 14

Core idea

Start hereEvery outcome with its chance

Toss three coins and count heads: $0$ , $1$ , $2$ or $3$ . Each count has a probability. Listing them all in a table gives a complete picture of the experiment: a probability distribution.

The big idea

A discrete random variable $X$ takes separate values $x$ , each with probability $P(X=x)$ . The table of values and probabilities is its distribution; the probabilities must sum to $1$ .

1. Building a distribution

Distribution of the total of two dice

From the 36-cell grid:

P(X=2)=\tfrac1{36}

, rising to

P(X=7)=\tfrac6{36}

, then falling. The bars add to

1

Worked exampleHeads in three tosses

Find the distribution of $X$ = number of heads.

Goal: Table.
Strategy: List the 8 equally likely outcomes.

$HHH;\ HHT,HTH,THH;\ HTT,THT,TTH;\ TTT$
Grouped by heads.
$P(0)=\tfrac18,\ P(1)=\tfrac38,\ P(2)=\tfrac38,\ P(3)=\tfrac18$
Count.

AnswerAs above.

✓ CheckSum

=1

✓.

Worked example $P(X=x)=kx$ for $x=1,2,3,4$

Find $k$ and $P(X\ge3)$ .

Goal: $k$ ; probability.
Strategy: Sum to $1$ .

$k(1+2+3+4)=1\Rightarrow k=\tfrac1{10}$
Total probability.
$P(X\ge3)=\tfrac{3+4}{10}=0.7$
Add.

Answer

k=0.1

;

0.7

✓ Check

0.1+0.2+0.3+0.4=1

✓.

Common misconception $<$ vs $\le$ For discrete

X

P(X<3)=P(X\le2)

. Read inequalities carefully.

Pause and think

Two cards from 2 red and 3 black without replacement. Distribution of reds?

Summary

List all values with probabilities.
$\sum P=1$ .

Random variable: A numerical outcome of a random process.

HistoryThe first probability textbook. In 1657 the Dutch scientist Christiaan Huygens published De ratiociniis in ludo aleae ("On reasoning in games of chance"), the first printed book on probability. It worked through dice and card problems by listing every possible payoff with its chance: probability distributions in all but name.

Fun factSpotting a loaded die. A fair die has the uniform distribution

P(X=x)=\tfrac16

. Casinos test dice by rolling them thousands of times and comparing the observed frequencies with this distribution; a die that shows six

20\%

of the time instead of

16.7\%

is quickly removed from play.

Section 3 of 14

Key formulae

Sum of probabilities

\sum_{\text{all }x}P(X=x)=1

Probabilities of ranges

P(X\ge a)=\sum_{x\ge a}P(X=x)=1-P(X<a)

TrickList the values before the probabilities. First write the top row of the table: every value

X

can take, smallest to largest. Then work out each probability, with a tree diagram if the experiment has stages. Finally check the bottom row adds to

1

; if it doesn't, a value or a branch is missing.

Section 4 of 14

Worked examples

Example 1 · A formula0 / 2 steps

P(X=x)=k(x+1)

for

x=0,1,2,3

. Find

k

and

P(X\ge2)

Example 2 · Without replacement0 / 3 steps

A bag has

3

red and

2

blue counters. Two are taken without replacement and

X

is the number of red counters. Find the distribution of

X

Example 3 · Two dice0 / 2 steps

Two fair dice are thrown and

D

is the (non-negative) difference between the scores. Find

P(D=0)

and

P(D=1)

Section 5 of 14

Explore the graph

A distribution given by a formula

Loading the interactive graph…

Try this

The bars show $P(X=x)=kx$ for $x=1,2,3,4$ .
For a valid distribution the bar heights must add to $1$ . Find the $k$ that does this.
For $k=0.2$ the heights add to $2$ . Why is that impossible?

A formula becomes a distribution only once $k$ makes the probabilities sum to $1$ .

In the real worldInsurance tables. Insurers model the number of claims a customer makes in a year as a random variable: perhaps

P(X=0)=0.85

P(X=1)=0.12

P(X=2)=0.03

. The whole premium calculation starts from a distribution table like this.

Section 6 of 14

Common mistakes

IncorrectLeaving out a value, such as

X=0

CorrectList every possible value

The probabilities then cannot sum to

1

Incorrect

P(X\ge2)

for values

0

–

3

=P(X=3)

Correct

P(X=2)+P(X=3)

"At least

2

" includes

2

IncorrectAssuming the values are equally likely

CorrectWork out each probability

Two heads and one head are not equally likely.

IncorrectA probability bigger than

1

or negative

CorrectCheck the table

Each probability must be in

[0,1]

Probabilities that add to $1.2$

Probabilities that add to exactly $1$

Section 7 of 14

Exam tip

Exam tipShow one probability in full. When asked to "show that

P(X=1)=\tfrac35

", write every factor of the calculation, including the "

\times2

" for the two orders. "Show that" questions give the answer, so the marks are entirely for the working.

Section 8 of 14

Guided practice

SA-9709-P5-DV1-0012 marksL1 FoundationCalculatorDistributions

The random variable

X

has

P(X=x)=k(x+1)

for

x=0,1,2,3

. Find

k

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-9709-P5-DV-0012 marksL1 FoundationCalculatorDistributions

The random variable

X

has

P(X=x)=kx

for

x=1,2,3,4

. Find

k

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P5-DV1-0021 markL1 FoundationCalculatorDistributions

X

takes the values

1

2

and

3

with probabilities

0.2

0.5

and

p

. Find

p

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P5-DV-0021 markL1 FoundationCalculatorBuilding tables

Three fair coins are thrown and

X

is the number of heads. Find

P(X=2)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P5-DV1-0051 markL1 FoundationCalculatorDistributions

X

takes the values

0,1,2,3

with probabilities

0.1,\ 0.3,\ 0.4,\ 0.2

. Find

P(X\ge2)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P5-DV1-0031 markL1 FoundationCalculatorBuilding tables

Two fair coins are tossed and

X

is the number of tails. Find

P(X=1)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P5-DV1-0043 marksL2 StandardCalculatorBuilding tables

A bag contains

3

red and

2

blue counters. Two are taken at random without replacement, and

X

is the number of red counters taken. Find

P(X=1)

and

P(X=2)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P5-DV1-0062 marksL2 StandardCalculatorBuilding tables

Two fair dice are thrown and

D

is the difference between the scores (larger minus smaller, or

0

if equal). Find

P(D=0)

and

P(D=1)

Salam Academy Original Practice · Proposed mark scheme

SA-9709-P5-DV1-0072 marksL2 StandardCalculatorDistributions

The random variable

X

has

P(X=x)=\dfrac kx

for

x=1,2,3

. Find

k

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-P5-DV-0135 marksL4 ChallengingCalculatorBuilding tables

Two fair dice are thrown and

X

is the larger of the two scores (or the common score if they are equal). Find

P(X=4)

and

E(X)

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

A discrete random variable takes separate values, each with a probability.
List every value, then every probability; they sum to $1$ .
Use the sum to find unknown constants.
Read ranges like $P(X\ge2)$ by adding the right columns.

Section 12 of 14

Quick quiz

Q1 · SA-9709-P5-DV-Q011 markL1 FoundationCalculatorDistributions

In any probability distribution, the probabilities

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-9709-P5-DV1-0081 markL1 FoundationCalculatorDistributions

Which of these could be a probability distribution for

X

taking the values

0,1,2

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 7357 A-level Mathematics
Where: Section O, Statistical distributions
Expected: Discrete random variables given in a table or by a simple formula.
Calculator: Permitted on all papers; a calculator with statistical distribution functions is expected.
Typical style: Questions may draw on the pre-released large data set.
Open in: 7357 · Paper 3 7367 · Paper 3

Section 14 of 14

Past paper questions

8 questions (48 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

9709/52 · Oct/Nov 2025Question 27 marks
Probability distribution from selections without replacement, and its variance.
Question paper, page 3 Mark scheme
9709/52 · May/June 2025Question 13 marks
Probability distribution for the number of tails from three biased coins.
Question paper, page 2 Mark scheme
9709/52 · Feb/March 2025Question 16 marks
Distribution of heads from three biased coins, then the variance.
Question paper, page 2 Mark scheme
9709/52 · Oct/Nov 2024Question 36 marks
Probability distribution of a coin-and-dice variable and its variance.
Question paper, page 5 Mark scheme
9709/52 · May/June 2024Question 57 marks
Probability distribution, expectation and variance of the total of two coins.
Question paper, page 8 Mark scheme
9709/52 · May/June 2023Question 16 marks
Probability distribution from a formula with an unknown constant, then mean and variance.
Question paper, page 3 Mark scheme

Show 2 more

9709/52 · Oct/Nov 2022Question 57 marks
Fair and biased coins: a conditional probability and a probability distribution.
Question paper, page 8 Mark scheme
9709/52 · May/June 2022Question 26 marks
Distribution of the sum of two rolls of a non-standard die, mean and variance.
Question paper, page 4 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).