Convergence of relative frequency
A conceptual Edexcel topic — short on calculation, heavy on reasoning. Tests understanding that more trials = better estimate.
The principle
As the number of trials in an experiment increases, the relative frequency of an outcome tends towards the true (theoretical) probability.
Formally: if outcome A has true probability p, and we run n trials with frequency f, then f/n → p as n → ∞.
Why we care
This justifies using large samples:
- A small sample (10–50 trials) gives a rough estimate.
- A medium sample (100–500) gives a usable estimate.
- A large sample (5000+) gives a very accurate estimate.
✦Worked example
A drawing pin is thrown 100 times. It lands point-up 67 times. Estimated P(point up) ≈ 0.67. The same pin is thrown 10 000 times. It lands point-up 6543 times. Now estimated P ≈ 0.6543.
The second estimate is more reliable.
Edexcel exam wording
Typical phrasing:
- "Sara thinks the dice is biased. She rolled it 30 times and got 9 sixes. Is this evidence the dice is biased? Explain."
- Expected (if fair) = 5; observed 9 — higher. But sample size 30 is small; deviation is plausible. Conclusion: insufficient evidence.
- "How could Sara improve her experiment?"
- Roll the dice many more times.
Edexcel exam tip
When asked "is this evidence of bias?" or "how could the experiment be improved?", Edexcel mark schemes consistently award:
- "Increase the number of trials" or "use a larger sample" B1.
- "Compare to expected number" B1.
- "Use a confidence interval / standard error" — A-Level level, not GCSE.
⚠Common mistakes— Common errors
- Treating relative frequency from a tiny sample (5–20 trials) as exact.
- Concluding bias from a small deviation in a small sample.
- Failing to reference sample size when criticising or improving an experiment.
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