Estimate probability and expected frequency
Use experimental relative frequency to estimate probability, and multiply a probability by the number of trials for an expected count.
Section 9 · Probability
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
How do you calculate relative frequency?
What you will learn
- Calculate relative frequency from repeated trials
- Interpret fair, biased and random outcomes
- Find expected frequencies and explain why they are not guaranteed counts
Relative frequency is evidence, not certainty
Relative frequency = observed number of times an event occurs ÷ number of trials. It can estimate the underlying probability, especially as a fair experiment is repeated many times.
Short runs vary by chance. A result such as 12 successes in 20 trials gives an estimate 0.6, but it does not prove that the next trial must succeed with probability exactly 0.6.
Expected frequency is a long-run average
Expected frequency = probability × number of trials. It may be fractional; actual outcomes are whole counts and can differ from the expectation.
A fair device has outcomes with the stated equal chances; a biased device does not. Random means an individual outcome is uncertain under the model, not that every short sample must match the expected proportion.
Pause and explain
A model gives P(success) = 0.2 over 80 trials. What is the expected success count?
Worked example
A spinner lands on yellow 36 times in 60 spins. Estimate the probability of yellow and the expected number of yellows in 250 further spins using that estimate.
Show the worked solution
- Relative frequency = 36/60 = 0.6.
- Use 0.6 as the estimated probability.
- Expected count = 250 × 0.6 = 150.
Answer Estimated probability 0.6; expected yellow results 150.
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 lands heads 47 times in 100 tosses. Give the relative frequency and estimated tails probability.
Give me a hint
Use the observed heads proportion, then take its complement.
Compare my reasoning
- Heads relative frequency = 47/100 = 0.47.
- Estimated tails probability = 1 − 0.47.
- The estimate is 0.53.
Heads 0.47; estimated tails 0.53.
Look for these in your work
- Observed proportion calculated
- Complement used
- Estimate labelled
A seed has an estimated 0.84 probability of germinating. In 350 seeds, find the expected number that germinate and the expected number that do not.
Give me a hint
Multiply each complementary probability by 350.
Compare my reasoning
- Expected germinations = 350 × 0.84 = 294.
- Non-germination probability = 0.16.
- Expected non-germinations = 350 × 0.16 = 56.
294 expected to germinate; 56 expected not to germinate.
Look for these in your work
- Probabilities total 1
- Both expected counts calculated
- Counts add to the sample size
A machine is tested in batches. It produces 18 defective parts in 600 trials. Using this rate, estimate the expected defective count in a production run of 2,500 parts and explain one limitation.
Give me a hint
Estimate with the observed relative frequency, then multiply by the larger run size.
Compare my reasoning
- Estimated defect probability = 18/600 = 0.03.
- Expected defects = 2500 × 0.03 = 75.
- This assumes conditions and machine performance stay similar; random runs can differ from 75.
About 75 expected defects, assuming the observed rate remains representative.
Look for these in your work
- Rate estimated
- Expected count calculated
- Assumption and variation stated
Common mistakes
- Calling an expected count a guaranteed result
- Using total outcomes instead of trial count for relative frequency
- Assuming a short run proves a device is fair
- Confusing random variation with bias
What can you explain now?
An event occurs 15 times in 50 trials. What is its relative frequency?
Compare with the explanation
0.3
15 ÷ 50 = 0.3.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why can the actual frequency differ from the expected frequency in a random experiment?
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.