Relative frequency is a key probability concept tested on both Foundation and Higher GCSE Maths papers across AQA, Edexcel and OCR. When you cannot calculate a theoretical probability — for example because a dice might be biased or a spinner's sections are not equal — you use experimental results instead. Relative frequency provides an estimate of probability based on observed data, and the more trials you carry out, the more reliable it becomes. This guide explains the concept, method and common exam questions. For an overview of every topic, see our complete GCSE Maths topics list.
What Is Relative Frequency?
Relative frequency (also called experimental probability) estimates the probability of an event based on the results of an experiment or collected data.
Key Formulas
Unlike theoretical probability, which uses equally likely outcomes, relative frequency is based on what actually happened.
The Law of Large Numbers
As the number of trials increases, the relative frequency of an event tends to get closer to the theoretical probability (if one exists). This is why more trials give a more reliable estimate.
Step-by-Step Method
- Read or carry out the experiment.
- Count how many times the event of interest occurred.
- Count the total number of trials.
- Divide: relative frequency = event occurrences ÷ total trials.
- State that this is an estimate of probability.
- If asked to improve the estimate, say that more trials should be carried out.
Worked Example 1 — Foundation Level
Question: A drawing pin is dropped 50 times. It lands point-up 18 times. Estimate the probability that it lands point-up.
Working:
Relative frequency = 18 ÷ 50 = 0.36.
Answer: The estimated probability of landing point-up is 0.36.
Worked Example 2 — Higher Level
Question: A biased dice is rolled. The table shows cumulative results.
| Total rolls | 50 | 100 | 200 | 500 |
|---|---|---|---|---|
| Times showing 4 | 12 | 21 | 38 | 84 |
(a) Calculate the relative frequency of rolling a 4 after each set of trials. (b) Which estimate is most reliable? (c) Estimate the probability of rolling a 4.
Working:
(a) After 50: 12/50 = 0.24. After 100: 21/100 = 0.21. After 200: 38/200 = 0.19. After 500: 84/500 = 0.168.
(b) The estimate after 500 trials is the most reliable because relative frequency becomes more accurate as the number of trials increases.
(c) The best estimate is 0.168, so P(4) is approximately 0.17 (2 d.p.).
Answer: (a) 0.24, 0.21, 0.19, 0.168. (b) 500 trials is most reliable. (c) Approximately 0.17.
Worked Example 3 — Exam Style
Question: A coin is flipped 400 times and lands on heads 212 times. Is the coin fair? Explain.
Working:
Relative frequency of heads = 212 ÷ 400 = 0.53.
A fair coin has P(heads) = 0.5. The relative frequency of 0.53 is close to 0.5. Over 400 trials, a difference of 0.03 is small and could be due to random variation.
Answer: The coin appears to be approximately fair. The relative frequency of 0.53 is close to the expected 0.5, and the small difference is consistent with natural variation over 400 trials.
Common Mistakes
- Calling relative frequency exact. It is always an estimate — use this word in your answer.
- Using too few trials to draw conclusions. A small number of trials produces unreliable estimates. Always mention that more trials would improve accuracy.
- Confusing relative frequency with theoretical probability. Relative frequency is based on experimental results; theoretical probability is calculated from equally likely outcomes.
Exam Tips
- Always use the word "estimate" when referring to relative frequency — this shows the examiner you understand the concept.
- If a table shows relative frequencies for increasing numbers of trials, the most reliable estimate is always the one with the most trials.
- To predict future outcomes, multiply the relative frequency by the number of future trials.
- Questions often combine relative frequency with expected frequency — see expected frequency.
- For foundational probability, see probability scale and basic probability. For key formulas, visit our GCSE Maths formulas page.
Practice Questions
Q1 (Foundation): A spinner is spun 80 times and lands on green 24 times. Estimate the probability of landing on green.
Q2 (Foundation): Explain how the estimate in Q1 could be made more reliable.
Q3 (Higher): A dice is rolled 1000 times. The number 5 appears 200 times. (a) Estimate P(5). (b) If the dice were fair, what would you expect? (c) Is the dice fair?
Practise relative frequency and experimental probability free on GCSEMathsAI.
Related Topics
Summary
- Relative frequency estimates probability from experimental results: event occurrences divided by total trials.
- It is always an estimate — use this word in exam answers.
- More trials produce a more reliable estimate because relative frequency tends towards the theoretical probability as the number of trials increases.
- Compare relative frequency with theoretical probability to assess whether an object (dice, coin, spinner) is fair.
- To make predictions, multiply the relative frequency by the number of future trials.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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