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Contents · 0% complete
  1. Learning objectives
  2. Core idea
  3. Key formulae
  4. Worked examples
  5. Explore the graph
  6. Common mistakes
  7. Exam tip
  8. Guided practice
  9. Practice questions
  10. Challenge
  11. Summary
  12. Quick quiz
  13. Same topic, different board
  14. Past paper questions
Learn8300Papers 1–3 · Higher tierProbabilitySingle events and relative frequency
Probability · Subtopic 1 of 4

Single Events and Relative Frequency

When outcomes are equally likely, probability =favourable outcomestotal outcomes=\dfrac{\text{favourable outcomes}}{\text{total outcomes}}. The probabilities of all outcomes add to 11, so P(not A)=1−P(A)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(not A)=1−P(A)P(\text{not }A)=1-P(A) and the fact that probabilities sum to 11.
  • Estimate probabilities with relative frequency.
  • Calculate expected frequency.
Pause and think

A coin is tossed 1010 times and lands heads 77 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=favourabletotalP=\frac{\text{favourable}}{\text{total}}. P(not A)=1−P(A)P(\text{not }A)=1-P(A). Relative frequency estimates PP; expected frequency =P×n=P\times n.

1. The scale

The probability scale
0.20.40.60.810.511.520probabilityimpossibleroll a 6 (1/6)heads (1/2)not a 6 (5/6)certain
P(not A)=1−P(A)P(\text{not }A)=1-P(A): rolling "not a 6" is 1−16=561-\tfrac16=\tfrac56.
Worked exampleA bag of 3 red, 5 blue, 2 green

P(blue); P(not green).

Goal
Probabilities.
Strategy
Count; complement.
  1. 510=12\tfrac5{10}=\tfrac12
    5 of 10.
  2. 1−210=8101-\tfrac2{10}=\tfrac8{10}
    Complement.
Answer12\frac12; 45\frac45.
✓ Check3+5=83+5=8 non-green ✓.

2. Relative frequency

Relative frequency of heads settles near 0.5 as tosses increase
501001502000.20.40.60.810number of tossesrelative frequencytheoretical 0.5
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.
  1. 36120=0.3\tfrac{36}{120}=0.3
    Estimate.
  2. 0.3×500=1500.3\times500=150
    Expected frequency.
Answer0.30.3; 150150.
✓ 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−P1-P.
  • More trials → better estimate.
Key words
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 00024\,000 times around 1900. He got 12 01212\,012 heads: a relative frequency of 0.50050.5005, extremely close to 12\tfrac12.
Section 3 of 14

Key formulae

Equally likely outcomesP(A)=favourabletotalP(A)=\frac{\text{favourable}}{\text{total}}
ComplementP(not A)=1−P(A)P(\text{not }A)=1-P(A)
Expected frequencyexpected number=P(A)×n\text{expected number}=P(A)\times n
TrickSanity check. Any probability above 11 or below 00 is wrong, and expected frequencies should be sensible: with P=0.15P=0.15 over 8080 games you expect about one win in seven, so 1212 is believable and 120120 is not.
Section 4 of 14

Worked examples

Example 1 · Missing probability0 / 1 steps
A spinner has outcomes 1,2,3,41,2,3,4 with probabilities 0.20.2, 0.30.3, xx and 0.10.1. Find xx.
    Example 2 · Relative frequency0 / 2 steps
    4646 of 200200 cars passing a school were red. Estimate the probability that the next car is red, and how many of the next 500500 will be red.
      Section 5 of 14

      Explore the graph

      Expected frequency
      Loading the interactive graph…
      Try this
      1. The line shows how many successes you expect in xx trials when the probability is pp.
      2. The point marks 8080 trials. Set p=0.15p=0.15: you expect 1212.
      3. What does the line look like when p=1p=1? When p=0p=0?

      Expected frequency grows in proportion to the number of trials.

      In the real world"70% chance of rain". A forecast of 70%70\% rain means that on days with conditions like today, it rained about 7070 times in 100100. It is a relative frequency from past weather and computer models, not a promise.
      Section 6 of 14

      Common mistakes

      IncorrectWriting "33 in 1010" or 3:103:10
      Correct310\tfrac{3}{10} or 0.30.3
      Probabilities are fractions, decimals or percentages.
      IncorrectUsing 16\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 11
      CorrectThey sum to exactly 11
      Check your arithmetic.

      I flipped a coin 10 times and got 7 heads

      Relative frequency 0.70.7. Fair coin, small sample.

      Section 7 of 14

      Exam tip

      Exam tipLeave fractions unsimplified if unsure. An unsimplified correct fraction like 610\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 66 red, 33 blue and 11 green counter. One counter is taken at random. Find the probability that it is not red.

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      Salam Academy Original Practice · Proposed mark scheme
      Section 9 of 14

      Practice questions

      SA-4024-PR1-0011 markL1 FoundationNon-calculatorSingle events
      A bag contains 55 red, 33 blue and 22 green counters. One is taken at random. Find the probability that it is blue.

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      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.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-PR-0011 markL1 FoundationNon-calculatorSingle events
      The probability that a train is late is 0.350.35. Find the probability that it is not late.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-PR-0111 markL1 FoundationNon-calculatorSingle events
      A biased spinner has outcomes 11, 22, 33, 44 with probabilities 0.20.2, 0.30.3, xx and 0.10.1. Find xx.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-PR1-0031 markL1 FoundationNon-calculatorFrequency
      A fair coin is tossed 5050 times. How many heads would you expect?

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-PR-0031 markL1 FoundationNon-calculatorFrequency
      The probability of winning a game is 0.150.15. How many wins would you expect in 8080 games?

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-PR1-0042 marksL2 StandardNon-calculatorFrequency
      Of 200200 cars passing a school, 4646 were red. (a) Estimate the probability that the next car is red. (b) Estimate how many of the next 500500 cars will be red.

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      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 120120 times and lands on six 2727 times. (a) Find the relative frequency of a six. (b) Use it to estimate how many sixes there would be in 600600 rolls.

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      Salam Academy Original Practice · Proposed mark scheme
      Section 11 of 14

      Summary

      • 0≤P(A)≤10\le P(A)\le1.
      • Equally likely: favourable ÷ total.
      • P(not A)=1−P(A)P(\text{not }A)=1-P(A); all probabilities sum to 11.
      • Relative frequency estimates PP; expected number =P×n=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?
      Choose one option

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      Salam Academy Original Practice · Proposed mark scheme
      Q2 · SA-4024-PR-Q021 markL1 FoundationCalculatorSingle events
      A dice is described as fair. This means
      Choose one option

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      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?
      Choose one option

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      Salam Academy Original Practice · Proposed mark scheme
      Section 13 of 14

      Same topic, different board

      Specification
      GCSE Mathematics 8300
      Where
      Probability
      Expected
      The same ideas; AQA also asks about fairness and sample size.
      Calculator
      Paper 1 non-calculator; Papers 2 and 3 calculator.
      Typical style
      Problem-solving questions often combine topics.
      Open in
      8300 · Papers 1–38300 · Papers 1–3
      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.

      1. 4024/22 · Oct/Nov 2025Question 112 marks

        Estimate faulty items in a batch from a sample.

        Question paper, page 7Mark scheme
      2. 4024/12 · May/June 2025Question 22 marks

        Probability of picking a ball from a bag.

        Question paper, page 3Mark scheme
      3. 4024/22 · May/June 2025Question 63 marks

        Relative frequency, and an expected count from a sample size.

        Question paper, page 5Mark scheme
      4. 4024/12 · May/June 2024Question 133 marks

        Relative frequency and expected frequency.

        Question paper, page 10Mark scheme

      All 4024 past paper questions by topic

      Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 8300, First teaching 2015 (official specification).