A coin is flipped 20 times and lands on heads 12 times. What is the relative frequency of heads?
Frequency ÷ total trials.
Relative frequency = 12 ÷ 20.
Relative Frequency
Relative frequency estimates probability based on experimental results. It builds on basic probability and links theoretical probability with real-world data.
Overview
Relative frequency is an estimate of probability based on actual results from an experiment.
\( \text{Relative frequency} = \frac{\text{number of successful outcomes}}{\text{total number of trials}} \)
It is sometimes called experimental probability.
Instead of using what should happen in theory, you use what actually happened.
What you should understand after this topic
- Understand what relative frequency means
- Calculate relative frequency from results
- Understand how it differs from theoretical probability
- Understand why more trials usually make results more reliable
- Use relative frequency to predict future outcomes
Key Definitions
Relative Frequency
An estimate of probability based on experimental results.
Experimental Probability
Another name for relative frequency.
Theoretical Probability
The probability expected from known equally likely outcomes.
Trial
One repeat of an experiment.
Outcome
The result of a trial.
Estimate
A value based on observed data, not exact certainty.
Key Rules
Main formula
\( \frac{\text{successes}}{\text{total trials}} \)
Use actual data
Relative frequency comes from results, not theory.
More trials
Usually give a more reliable estimate.
Can predict future results
You can use relative frequency to estimate later outcomes.
How to Solve
Step 1: Understand relative frequency
Relative frequency estimates probability using experimental results.
\( \text{Relative frequency} = \frac{\text{frequency of event}}{\text{total number of trials}} \)
Exam tip: Relative frequency comes from real data, not theoretical probability.
This builds on basic probability.
Step 2: Identify the event and total trials
Read the question carefully to find the event and the total number of trials.
Event frequency = how many times the event happened.
Total trials = how many times the experiment was repeated.
Step 3: Divide to estimate probability
\( \frac{22}{50} = 0.44 \)
This means the event happened 44% of the time.
Step 4: Compare with theoretical probability
Theoretical probability
Uses expected outcomes, such as \(\frac{1}{2}\) for heads on a fair coin. See probability basics.
Relative frequency
Uses actual experimental results.
Step 5: Understand why more trials matter
With more trials, relative frequency usually becomes a better estimate.
Exam thinking: More trials usually give a more reliable estimate.
Step 6: Predict future outcomes
Use relative frequency to estimate future results.
Expected number of outcomes = relative frequency × number of future trials
Step 7: Exam method summary
See probability basics for probability notation.
- Identify the event.
- Find the frequency of that event.
- Find the total number of trials.
- Divide event frequency by total trials.
- Use the result to estimate probability or predict outcomes.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics, focusing on calculating relative frequency from experiments.
Edexcel
A coin is flipped 40 times. It lands on heads 17 times.
Find the relative frequency of heads.
Edexcel
A spinner is spun 30 times. It lands on blue 9 times.
Estimate the probability of landing on blue.
Edexcel
A dice is rolled 80 times. An even number occurs 46 times.
Find the relative frequency of getting an even number.
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on using relative frequency to make predictions.
AQA
A biased coin is flipped 90 times. It lands on tails 63 times.
Estimate how many times it will land on tails in 300 flips.
AQA
A machine produces bolts. In a test, 12 out of 50 bolts are faulty.
Estimate how many faulty bolts there will be in a batch of 200.
AQA
A student records the results of an experiment.
They say the relative frequency is a good estimate of probability.
Tick one box. Always ☐ Sometimes ☐ Never ☐
Give a reason for your answer.
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising interpretation and reliability of relative frequency.
OCR
An experiment is repeated many times. The relative frequency of an event is 0.35.
Estimate how many times the event will occur in 120 trials.
OCR
Two experiments are carried out.
Experiment A uses 10 trials. Experiment B uses 1000 trials.
Which experiment is likely to give a more reliable estimate of probability?
OCR
Explain why relative frequency tends to get closer to the true probability as the number of trials increases.
Exam Checklist
Step 1
Find the number of times the event happened.
Step 2
Find the total number of trials.
Step 3
Use frequency ÷ total trials.
Step 4
If predicting future outcomes, multiply the relative frequency by the new number of trials.
Most common exam mistakes
Wrong order
Writing total ÷ frequency instead of frequency ÷ total.
Theory confusion
Using theoretical probability when the question wants relative frequency.
Prediction mistake
Forgetting to multiply by the new number of trials.
Reliability mistake
Not recognising that larger samples are usually more reliable.
Common Mistakes
These are common mistakes students make when working with relative frequency in GCSE Maths.
Reversing the fraction
Incorrect
A student writes total ÷ frequency instead of frequency ÷ total.
Correct
Relative frequency is calculated as \(\frac{\text{number of successes}}{\text{total number of trials}}\).
Using theoretical instead of experimental values
Incorrect
A student ignores the data and uses expected probabilities.
Correct
Relative frequency must be based on experimental results given in the question, not theoretical probabilities.
Not scaling for predictions
Incorrect
A student uses the relative frequency directly without adjusting for a new number of trials.
Correct
To predict outcomes, multiply the relative frequency by the new total number of trials.
Reading the wrong value from data
Incorrect
A student selects an incorrect frequency from a table or chart.
Correct
Carefully identify the correct category and corresponding frequency before calculating.
Expecting exact agreement with theory
Incorrect
A student assumes relative frequency must match theoretical probability exactly.
Correct
Relative frequency is an estimate. It approaches the theoretical probability as the number of trials increases.
Try It Yourself
Practise estimating probabilities using relative frequency.
Foundation
Foundation Practice
Estimate probabilities using relative frequency.
A die is rolled 50 times. A 6 appears 10 times. Estimate the probability of rolling a 6.
Use frequency ÷ trials.
10 ÷ 50 = 1/5.
What is relative frequency?
Think proportion.
Relative frequency = successes ÷ total trials.
A spinner is spun 40 times and lands on A 16 times. Estimate probability of A.
Simplify 16/40.
16 ÷ 40 = 2/5.
If the number of trials increases, what happens to relative frequency?
Think accuracy.
More trials → more reliable estimate.
A coin is flipped 100 times and lands on heads 55 times. Estimate probability of heads.
Use decimal.
55 ÷ 100 = 0.55.
A student says relative frequency is always exactly equal to theoretical probability. What is wrong?
Think randomness.
Relative frequency estimates probability.
A die is rolled 30 times. A 4 appears 6 times. Estimate probability of rolling a 4.
6 ÷ 30.
6 ÷ 30 = 1/5.
Which value is a relative frequency?
Must be between 0 and 1.
Relative frequency is a probability value.
A spinner lands on blue 25 times out of 50. Estimate probability of blue.
Simplify fraction.
25 ÷ 50 = 1/2.
Higher
Higher Practice
Use relative frequency to make predictions and compare probabilities.
A biased coin is flipped 200 times and lands on heads 120 times. Estimate probability of heads.
120 ÷ 200.
120 ÷ 200 = 0.6.
A spinner is spun 80 times and lands on red 20 times. Estimate probability of red.
Simplify 20/80.
20 ÷ 80 = 1/4.
A student uses 5 trials to estimate probability. What is the issue?
Think reliability.
Small sample → unreliable estimate.
A die is rolled 120 times and a 6 appears 18 times. Estimate probability of 6.
Simplify 18/120.
18 ÷ 120 = 3/20.
What happens to relative frequency as trials approach infinity?
Law of large numbers.
Relative frequency approaches true probability.
A spinner lands on green 45 times out of 150 spins. Estimate probability.
Simplify fraction.
45 ÷ 150 = 3/10.
A relative frequency is 0.48 after many trials. What does this suggest?
Estimate only.
It is an estimate, not exact.
A machine produces defective items. 12 out of 300 are defective. Estimate probability of defect.
Simplify 12/300.
12 ÷ 300 = 1/25.
Why might relative frequency differ from theoretical probability?
Think randomness.
Random variation causes differences.
A coin is flipped 500 times and lands heads 260 times. Estimate probability of heads.
Simplify 260/500.
260 ÷ 500 = 13/25.
Games
Practise this topic with interactive games.
Games coming soon.
Frequently Asked Questions
What is relative frequency?
Experimental probability.
How is it calculated?
Number of successes divided by trials.
What happens with more trials?
It approaches theoretical probability.