Relative frequency and expected outcomes
Estimate probability from repeated trials and use it to predict a frequency without promising an exact result.
Section C8.2 / E8.2 · Relative and expected frequencies
Core and Extended. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Divide frequencies by totals.
- Multiply a count by a fraction or decimal.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What denominator is used in relative frequency?
What you will learn
- Estimate probability using relative frequency.
- Calculate expected frequency and discuss reliability or bias.
Observed proportions estimate probability
Relative frequency is the number of event occurrences divided by the number of trials. If a spinner lands on blue 18 times in 60 spins, its estimated blue probability is 18/60 = 0.3. This estimate reflects these observations rather than proving an exact underlying probability.
A larger number of fair, independent trials generally gives a more stable estimate, but it does not repair biased sampling or a changed experiment. A fair die should have approximately equal long-run face frequencies; a short uneven run alone does not prove that the die is biased.
Expected frequency is a long-run prediction
Multiply probability by the number of future trials. With estimated probability 0.3, the expected number of blue outcomes in 200 comparable spins is 200 × 0.3 = 60. Actual results can differ; expected does not mean guaranteed.
An expected frequency can be non-integer, such as 7.5 for fifteen trials at probability one half. It is an average prediction across many repeated sets, not a claim that half an event occurs in one set. State the assumption that the future trial conditions resemble those used for the estimate.
Pause and explain
Estimated P(blue) = 0.3. How many blue outcomes are expected in 200 trials?
Worked example
Blue appears 18 times in 60 spins. Estimate blue occurrences in 200 further comparable spins.
Show the worked solution
- Estimate P(blue) = 18/60 = 0.3.
- Expected frequency is 200 × 0.3 = 60.
- Sixty is a prediction; the actual future count is not guaranteed.
Answer 60 expected blue outcomes.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A coin gives 24 heads in 50 tosses. Estimate P(heads).
Give me a hint
Use heads divided by total tosses.
Compare my reasoning
- The observed proportion is 24/50.
- This is 0.48.
- It estimates probability for comparable tosses, rather than proving the coin biased.
0.48.
Look for these in your work
- Used relative frequency.
- Separated estimate from certainty.
An event occurs 35 times in 100 trials. Predict its frequency in 400 similar trials.
Give me a hint
Find the proportion before multiplying.
Compare my reasoning
- Estimated probability is 35/100 = 0.35.
- Expected count is 400 × 0.35 = 140.
- The actual count may differ from this expectation.
140 expected occurrences.
Look for these in your work
- Used the correct trial total.
- Explained the prediction limit.
A website polls only its most active users about a study habit. Will increasing the same biased sample automatically make it representative?
Give me a hint
Trial count and sample selection are separate issues.
Compare my reasoning
- The active-user group can differ from other learners.
- More observations from the same selected group can preserve the bias.
- A representative selection method is needed as well as a sufficient sample size.
No; the sampling method also needs to be representative.
Look for these in your work
- Identified selection bias.
- Avoided treating sample size as a complete remedy.
Common mistakes
- Treating expectation as a guarantee.
- Claiming a short uneven run proves bias.
- Ignoring whether future trials are comparable.
What can you explain now?
Must an expected frequency be an integer?
Compare with the explanation
No.
It is a long-run average prediction, so a fractional expectation is meaningful even though an actual count is whole.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, turn an observed frequency into a future prediction and state one assumption.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.