Estimate probability from experimental relative frequency
Divide event count by trial count · Combine experiments using counts · Distinguish an estimate from the underlying probability
Section Paper 1 · Historical Core · non-calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Fractions and proportions
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Divide event count by trial count
- Combine experiments using counts
- Distinguish an estimate from the underlying probability
Count divided by trials
Relative frequency is event count divided by the total number of comparable trials. Eleven heads in twenty tosses gives 11/20=0.55, an estimate of the head probability. It does not prove the coin has probability exactly 0.55. A small experiment can fluctuate greatly, as the illustrated early cumulative values show.
More independent, representative trials generally provide a more stable estimate, but do not guarantee monotonic improvement or agreement with the true probability. Changing the mechanism or pooling incompatible conditions can make an estimate misleading.
Pool counts, not unweighted percentages
If one experiment has 3 successes in 10 trials and another 45 in 90, the pooled estimate is 48/100=0.48. Averaging 30% and 50% to get 40% gives the ten-trial experiment too much weight. A zero observed count does not prove that an event is impossible.
Sources for this lesson
Pause and explain
Do zero red outcomes in ten trials prove P(red)=0?
Worked example
A spinner lands on blue 18 times in 60 spins. Estimate P(blue).
Show the worked solution
- Use observed blue count over total spins.
- 18/60=3/10.
Answer 0.3 or 30%, an experimental estimate.
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 counter is drawn and replaced 50 times, with red observed 17 times. Estimate P(red).
Give me a hint
Divide red count by trial count.
Compare my reasoning
- There are 17 successes out of 50 trials.
- 17/50=0.34.
0.34 or 34%, an estimate.
Look for these in your work
- Use the correct denominator
- Label the result as estimated
Under unchanged conditions, two experiments give 3 successes in 10 trials and 45 in 90. Find the pooled relative frequency.
Give me a hint
Add counts and trials separately.
Compare my reasoning
- Total successes=48; total trials=100.
- 48/100=0.48, rather than an unweighted mean of the two percentages.
0.48 or 48%.
Look for these in your work
- Weight by numbers of trials
- Pool comparable conditions
An experiment gives 7 heads in 10 tosses; another gives 52 in 100. Which usually provides a more stable estimate under the same independent-toss model? Does its estimate have to be closer to the true probability?
Give me a hint
Separate a typical advantage from a guarantee.
Compare my reasoning
- The 100-toss sample usually has less sampling variability.
- Any particular finite sample can still be farther from the true probability.
The 100-toss sample is usually more stable; closeness is not guaranteed.
Look for these in your work
- Recognise sample-size effects
- Avoid a deterministic claim
Common mistakes
- Use the total trial count as denominator.
- Do not treat a sample proportion as exact theoretical probability.
What can you explain now?
Do zero red outcomes in ten trials prove P(red)=0?
Compare with the explanation
No; an event with positive probability can be absent in a small sample.
Observed frequency and underlying probability are different quantities.
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Make it stick
Do zero red outcomes in ten trials prove P(red)=0?
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.