Calculate expected occurrences and interpret a fractional count
Multiply trial count by event probability · Interpret expected count as a long-run average · Keep expected and observed counts distinct
Section Paper 4 · Historical Extended · 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
- Probability estimates
- Multiplication with fractions
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
- Multiply trial count by event probability
- Interpret expected count as a long-run average
- Keep expected and observed counts distinct
An average prediction
If each of n trials has event probability p, the expected number of occurrences is np. For 60 draws with replacement and P(red)=1/4, the expected count is 15. This is an average across many comparable sixty-draw experiments, not a guarantee that this experiment gives 15.
Independence is useful for a stable repeated-trial model, but the expectation np only needs each trial to have probability p. If trial probabilities differ, add them instead. Do not apply one fixed p to a changing mechanism without justification.
A fractional expectation is meaningful
Ten rolls of a fair die have expected sixes 10/6=5/3. An actual count must be a whole number, but an average need not be. Preserve the exact fraction or requested precision. If p was estimated experimentally, np is an estimated expected count; its uncertainty does not disappear when multiplied.
Sources for this lesson
Pause and explain
Must an expected count be a whole number?
Worked example
For 80 independent trials with event probability 0.35, find the expected count.
Show the worked solution
- Expected count=80×0.35.
- Interpret the result as a repeated-experiment average.
Answer 28 occurrences on average, not a guaranteed count.
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 model gives success probability 3/8 on each of 120 trials. Find the expected successes.
Give me a hint
Multiply 120 by 3/8.
Compare my reasoning
- 120×3/8=45.
- This is an average prediction, not a promise.
45 successes on average.
Look for these in your work
- Use the probability per trial
- Interpret the count
Find the expected number of sixes in ten fair die rolls. Must an actual experiment give this result?
Give me a hint
Multiply ten by 1/6.
Compare my reasoning
- Expectation=10/6=5/3.
- An observed count is an integer and need not equal this fractional average.
5/3; no actual count equals 5/3.
Look for these in your work
- Keep the fraction
- Separate expectation from observation
An unchanged process gives 18 successes in 60 trials. Use this evidence to predict the expected successes in 250 future comparable trials. Is the result certain?
Give me a hint
Use the experimental estimate 18/60.
Compare my reasoning
- Estimated probability=0.3.
- Estimated expectation=250×0.3=75.
Estimated expectation 75; the observed count is not guaranteed.
Look for these in your work
- Carry the estimate qualification forward
- Use comparable conditions
Common mistakes
- Do not round an expectation just because observations are integers.
- An estimated probability gives an estimated expectation.
What can you explain now?
Must an expected count be a whole number?
Compare with the explanation
No; it is an average and can be fractional.
Individual counts are whole numbers, while their long-run average need not be.
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Make it stick
Must an expected count be a whole number?
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.