GCSE Maths · Probability

Relative frequency GCSE Questions and Worked Answers

Relative frequency is the number of observed successes divided by the total trials. It estimates probability when the experiment represents the situation you want to predict; it does not guarantee future counts.

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Start with the meaning

What you need to know about relative frequency

Spin a spinner 60 times and record blue on 18 spins. Blue occurred in 18 out of the 60 trials. That fraction, 18/60 = 0.3, describes what happened in this experiment. It is called relative frequency: the frequency compared with the total number of attempts.

See the idea first

Observed proportion, then a probability estimate

A trial is one attempt, such as one spin. An event is the outcome you are counting, such as blue. Relative frequency lies between 0 and 1. If the spinner and spinning conditions stay the same, the observed proportion can estimate the chance of blue on another spin.

One original spinner experiment
Colour	Count in 60 spins	Relative frequency
Blue	18	18/60 = 0.30
Red	27	27/60 = 0.45
Green	15	15/60 = 0.25

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Estimate a probability

What the problem asks: A drawing pin lands point-up 28 times in 80 drops. Estimate the chance of point-up.

How to solve it: 28 ÷ 80 = 0.35. This is an experimental estimate, not a claim that the true probability is exactly 0.35.

Predict a future count

What the problem asks: Using an estimate of 0.35, how many point-up results would you expect in 200 similar drops?

How to solve it: 0.35 × 200 = 70. Expect about 70, but the actual count can vary.

Combine experiments

What the problem asks: One run has 8 successes in 20 trials; another has 21 in 70. Find the overall relative frequency.

How to solve it: Add successes and trials separately: 29 ÷ 90, about 0.322. Do not take an unweighted mean of the two rates because the run sizes differ.

Judge reliability

What the problem asks: Which of 20 or 2000 similar unbiased trials generally gives a more reliable estimate?

How to solve it: The larger experiment usually has less random variation in its proportion. It does not correct bias or guarantee that every larger sample is closer to the true probability.

A reliable routine

Estimate future outcomes from observed trials

Use relative frequency when theoretical probabilities are unknown or you are assessing experimental results. Treating the observed proportion as a probability is reasonable only if the trial method is representative and future conditions are comparable.

Count the event occurrences and all relevant trials.
Divide occurrences by trials; check the result is between 0 and 1.
For a future expected count, multiply the estimate by the future number of trials.
State uncertainty and whether sample size, bias or changing conditions limit the conclusion.

Check: A fair coin need not give exactly half heads in a finite experiment. Larger samples tend to stabilise proportions, not make the running estimate improve after every toss.

Fully worked

Relative frequency GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Observed probability estimate

2 marks

Question

A drawing pin lands point-up 39 times in 100 drops. Estimate the probability of point-up.

\hat p=\frac{39}{100}=0.39

This is 39%. The hat on p simply indicates an estimated probability.

Example 2

The other outcome

2 marks

Question

A coin lands heads 47 times in 120 tosses. Find the relative frequency of tails.

Assuming every toss is recorded as heads or tails, tails count is $120-47=73$ .

\text{Relative frequency}=\frac{73}{120}

=0.60833\ldots

This is about 0.608, or 60.8%. Tip: use the event requested, not the count already given.

Example 3

Expected count

2 marks

Question

A component fails in 12 of 300 comparable tests. Estimate the number of failures in 2000 further tests.

\hat p=12/300=0.04

\text{Expected failures}=0.04(2000)=80

Expect about 80; do not promise exactly 80.

Example 4

Combine unequal runs

3 marks

Question

The same spinner gives red 9 times in 30 spins and 28 times in 70 more spins. Find red's overall relative frequency.

\text{Red total}=9+28=37

\text{Spin total}=30+70=100

\text{Relative frequency}=37/100=0.37

Weight runs by their sizes through their counts.

Example 5

Assess fairness

3 marks

Question

A die is rolled 60 times and gives six on 15 rolls. Does this prove the die is unfair?

Observed frequency is $15/60=0.25$ , compared with theoretical $1/6\approx0.167$ . The difference could arise from chance in a finite sample. It does not prove unfairness. More well-controlled trials could provide stronger evidence.

Example 6

More data, same bias

3 marks

Question

A survey of 5000 visitors to a sports shop finds 80% exercise weekly. Is this a reliable estimate for all adults?

Not necessarily. Sports-shop visitors may exercise more than adults generally. The large count reduces random variation within that sampling method but does not remove selection bias. Use a more representative sample.

10 original questions · total 23 marks

Relative frequency GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 28 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed

Relative frequency

2 marks

A spinner lands green 24 times in 80 spins. Find green's relative frequency.

Show worked answer

24/80=0.3

Green occurred in 30% of the trials.

Failure fraction

2 marks

Seven of 140 items fail a test. Estimate failure probability.

Show worked answer

\hat p=7/140=0.05

This is an experimental estimate of 5%.

Expected count

2 marks

An event has estimated probability 0.12. How many occurrences are expected in 250 similar trials?

Show worked answer

0.12(250)=30

Expect about 30, with random variation.

Complement

2 marks

A pin lands point-up 35 times in 100 drops. Estimate the probability it does not land point-up.

Show worked answer

There are $100-35=65$ other outcomes.

65/100=0.65

Recover a count

2 marks

In 200 trials an event has relative frequency 0.28. How many times did it occur?

Show worked answer

0.28(200)=56

Here 56 is the observed count, because the supplied rate describes those completed trials.

Combine counts

3 marks

An event occurs 6 times in 20 trials and 24 times in 80 further comparable trials. Find overall relative frequency.

Show worked answer

\frac{6+24}{20+80}=\frac{30}{100}=0.3

Add counts, then divide.

Unequal rates

3 marks

A has 4 successes in 10 trials; B has 18 in 90 comparable trials. Find the combined rate.

Show worked answer

\hat p=\frac{4+18}{10+90}=\frac{22}{100}=0.22

The simple mean of 0.4 and 0.2 would be 0.3, which ignores unequal trial counts.

Sample size

2 marks

Why are 1000 unbiased trials generally preferable to 10 unbiased trials for estimating a stable probability?

Show worked answer

The proportion usually fluctuates less with more observations, making it more reliable. However, it is not guaranteed to be closer in every particular experiment.

Changing conditions

2 marks

A machine's failure rate was estimated before a repair. Can you assume that estimate still applies afterwards?

Show worked answer

No. Repair may change the failure probability. Collect comparable post-repair evidence rather than assuming the old conditions still hold.

A running estimate

3 marks

A fair coin gives 5 heads in 10 tosses. The next toss is heads. Compare the two relative frequencies and explain what this shows.

Show worked answer

Initially $5/10=0.5$ . Afterwards $6/11\approx0.545$ . The estimate moved farther from the true 0.5 despite more data. Long-run stabilisation does not mean improvement after every trial.

Examiner-style feedback

Common relative frequency mistakes

Dividing the wrong way

Successes ÷ trials gives a proportion at most 1; trials ÷ successes usually does not.

Expected means guaranteed

An expected count is a model-based average, not a promise about one future experiment.

Large samples cure bias

Repeating an unrepresentative method more often does not make it representative.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Count the requested event.
Divide by total trials.
Multiply for an expected future count.
Check assumptions and uncertainty.

Quick answers

Relative frequency FAQ

How is theoretical probability different?

It comes from a model, such as equally likely faces of a fair die. Relative frequency comes from observed results.

Must an expected count be a whole number?

No. It is an average prediction; actual observed counts must be whole numbers, but expected values can be fractional.

Build connected skills

What to revise next

Organise observations

Frequency tables

Count outcomes clearly before dividing.

Revise frequency tables

Several stages

Tree diagrams

Combine probabilities with explicit assumptions.

Revise tree diagrams

Content standards

Curriculum and rights review

P1–P5 observed frequencies, probability estimates, expected outcomes and reliability across tiers. No formal hypothesis test is claimed. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references

AQA GCSE Maths subject content