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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

Learn4024Paper 1 · Non-calculatorProbabilitySingle events and relative frequency

Probability · Subtopic 1 of 3

Single Events and Relative Frequency

When outcomes are equally likely, probability $=\dfrac{\text{favourable outcomes}}{\text{total outcomes}}$ . The probabilities of all outcomes add to $1$ , so $P(\text{not }A)=1-P(A)$ . When you can't count outcomes, estimate the probability from experiments: relative frequency.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 25 minRevision 5 · updated 2026-10-27

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Section 1 of 14

Learning objectives

Find probabilities of equally likely outcomes.
Use $P(\text{not }A)=1-P(A)$ and the fact that probabilities sum to $1$ .
Estimate probabilities with relative frequency.
Calculate expected frequency.

Pause and think

A coin is tossed $10$ times and lands heads $7$ times. Is the coin biased?

Section 2 of 14

Core idea

Start hereHow likely is rain?

A forecast says 30% chance of rain. A coin has a 50% chance of heads. Probability puts every uncertain event on one scale, from 0 (impossible) to 1 (certain).

The big idea

Equally likely outcomes: $P=\frac{\text{favourable}}{\text{total}}$ . $P(\text{not }A)=1-P(A)$ . Relative frequency estimates $P$ ; expected frequency $=P\times n$ .

1. The scale

The probability scale

P(\text{not }A)=1-P(A)

: rolling "not a 6" is

1-\tfrac16=\tfrac56

Worked exampleA bag of 3 red, 5 blue, 2 green

P(blue); P(not green).

Goal: Probabilities.
Strategy: Count; complement.

$\tfrac5{10}=\tfrac12$
5 of 10.
$1-\tfrac2{10}=\tfrac8{10}$
Complement.

Answer

\frac12

;

\frac45

✓ Check

3+5=8

non-green ✓.

2. Relative frequency

Relative frequency of heads settles near 0.5 as tosses increase

Small experiments can be far off; more trials give a better estimate.

Worked exampleA drawing pin

Point up 36 times in 120 throws. Estimate; expected in 500.

Goal: Estimate; expectation.
Strategy: Relative frequency.

$\tfrac{36}{120}=0.3$
Estimate.
$0.3\times500=150$
Expected frequency.

Answer

0.3

;

150

✓ CheckMore throws give a better estimate ✓.

Common misconceptionExpected = guaranteed 150 is the long-run average, not a certainty.

Pause and think

P(rain) = 0.35. P(no rain)?

Summary

Favourable ÷ total.
Complement $1-P$ .
More trials → better estimate.

Relative frequency: Experimental probability.

HistoryA gambler's book. Gerolamo Cardano, an Italian doctor and keen gambler, wrote The Book on Games of Chance in the 1500s, one of the first works to calculate probabilities with dice. It was published only in 1663, nearly a century after his death.

Fun fact24,000 coin tosses. The statistician Karl Pearson tossed a coin

24\,000

times around 1900. He got

12\,012

heads: a relative frequency of

0.5005

, extremely close to

\tfrac12

Section 3 of 14

Key formulae

Equally likely outcomes

P(A)=\frac{\text{favourable}}{\text{total}}

Complement

P(\text{not }A)=1-P(A)

Expected frequency

\text{expected number}=P(A)\times n

TrickSanity check. Any probability above

1

or below

0

is wrong, and expected frequencies should be sensible: with

P=0.15

over

80

games you expect about one win in seven, so

12

is believable and

120

is not.

Section 4 of 14

Worked examples

Example 1 · Missing probability0 / 1 steps

A spinner has outcomes

1,2,3,4

with probabilities

0.2

0.3

x

and

0.1

. Find

x

Example 2 · Relative frequency0 / 2 steps

46

200

cars passing a school were red. Estimate the probability that the next car is red, and how many of the next

500

will be red.

Section 5 of 14

Explore the graph

Expected frequency

Loading the interactive graph…

Try this

The line shows how many successes you expect in $x$ trials when the probability is $p$ .
The point marks $80$ trials. Set $p=0.15$ : you expect $12$ .
What does the line look like when $p=1$ ? When $p=0$ ?

Expected frequency grows in proportion to the number of trials.

In the real world"70% chance of rain". A forecast of

70\%

rain means that on days with conditions like today, it rained about

70

times in

100

. It is a relative frequency from past weather and computer models, not a promise.

Section 6 of 14

Common mistakes

IncorrectWriting "

3

10

" or

3:10

Correct

\tfrac{3}{10}

0.3

Probabilities are fractions, decimals or percentages.

IncorrectUsing

\tfrac16

when experimental data are given

CorrectUse the relative frequency

The question may be about a biased dice.

IncorrectExpected frequency as a probability

CorrectIt is a number of times

Multiply the probability by the number of trials.

IncorrectProbabilities summing to more than

1

CorrectThey sum to exactly

1

Check your arithmetic.

I flipped a coin 10 times and got 7 heads

Relative frequency $0.7$ . Fair coin, small sample.

Section 7 of 14

Exam tip

Exam tipLeave fractions unsimplified if unsure. An unsimplified correct fraction like

\tfrac{6}{10}

usually earns full marks, but a wrongly simplified one loses them. Never give a probability as a ratio.

Section 8 of 14

Guided practice

SA-4024-PR-G011 markL1 FoundationNon-calculatorSingle events

A bag contains

6

red,

3

blue and

1

green counter. One counter is taken at random. Find the probability that it is not red.

Salam Academy Original Practice · Proposed mark scheme

Section 9 of 14

Practice questions

SA-4024-PR1-0011 markL1 FoundationNon-calculatorSingle events

A bag contains

5

red,

3

blue and

2

green counters. One is taken at random. Find the probability that it is blue.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-PR1-0021 markL1 FoundationNon-calculatorSingle events

A fair six-sided dice is rolled. Find the probability that the score is a prime number.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-PR-0011 markL1 FoundationNon-calculatorSingle events

The probability that a train is late is

0.35

. Find the probability that it is not late.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-PR-0111 markL1 FoundationNon-calculatorSingle events

A biased spinner has outcomes

1

2

3

4

with probabilities

0.2

0.3

x

and

0.1

. Find

x

Salam Academy Original Practice · Proposed mark scheme

SA-4024-PR1-0031 markL1 FoundationNon-calculatorFrequency

A fair coin is tossed

50

times. How many heads would you expect?

Salam Academy Original Practice · Proposed mark scheme

SA-4024-PR-0031 markL1 FoundationNon-calculatorFrequency

The probability of winning a game is

0.15

. How many wins would you expect in

80

games?

Salam Academy Original Practice · Proposed mark scheme

SA-4024-PR1-0042 marksL2 StandardNon-calculatorFrequency

200

cars passing a school,

46

were red. (a) Estimate the probability that the next car is red. (b) Estimate how many of the next

500

cars will be red.

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-4024-PR-0022 marksL2 StandardCalculatorFrequency

A dice is rolled

120

times and lands on six

27

times. (a) Find the relative frequency of a six. (b) Use it to estimate how many sixes there would be in

600

rolls.

Salam Academy Original Practice · Proposed mark scheme

Section 11 of 14

Summary

$0\le P(A)\le1$ .
Equally likely: favourable ÷ total.
$P(\text{not }A)=1-P(A)$ ; all probabilities sum to $1$ .
Relative frequency estimates $P$ ; expected number $=P\times n$ .

Section 12 of 14

Quick quiz

Q1 · SA-4024-PR-Q011 markL1 FoundationCalculatorSingle events

Which of these could NOT be a probability?

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-4024-PR-Q021 markL1 FoundationCalculatorSingle events

A dice is described as fair. This means

Salam Academy Original Practice · Proposed mark scheme

Q3 · SA-4024-PR1-0051 markL1 FoundationCalculatorFrequency

Four students each estimate the probability that a drawing pin lands point up. Whose estimate is most reliable?

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 4024 O Level Mathematics
Where: Papers 1 and 2 · 8.1–8.2 Probability; relative and expected frequency
Expected: Probability on a $0$ to $1$ scale; $P(\text{not }A)=1-P(A)$ ; probabilities summing to $1$ ; relative frequency as an estimate; expected frequency.
Calculator: Paper 1 is non-calculator; Paper 2 allows a calculator.
Typical style: A spinner was spun 120 times; estimate the probability of red and how many reds you would expect in 600 spins.
Open in: 4024 · Paper 1 4024 · Paper 2

Section 14 of 14

Past paper questions

4 questions (10 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

4024/22 · Oct/Nov 2025Question 112 marks
Estimate faulty items in a batch from a sample.
Question paper, page 7 Mark scheme
4024/12 · May/June 2025Question 22 marks
Probability of picking a ball from a bag.
Question paper, page 3 Mark scheme
4024/22 · May/June 2025Question 63 marks
Relative frequency, and an expected count from a sample size.
Question paper, page 5 Mark scheme
4024/12 · May/June 2024Question 133 marks
Relative frequency and expected frequency.
Question paper, page 10 Mark scheme

All 4024 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Cambridge International 4024, Current syllabus (official specification).