Empirical samples and the law of large numbers

OCR J560 Statistics expects qualitative understanding: as sample size grows, the relative frequency of outcomes gets close to their true probability.

Statement

If a random experiment is repeated n times and an event has true probability p, then the relative frequency of the event in those n trials approaches p as n → ∞.

This is the law of large numbers. It justifies using relative frequency as an estimate of probability.

What this looks like in practice

For a fair coin, P(heads) = 0.5.

Trials	Likely range of heads count
10	3 to 7 (very wide)
100	40 to 60
1000	470 to 530
10 000	4900 to 5100

The range as a proportion shrinks as n grows. Single trials fluctuate; long runs stabilise.

Implications for sampling

• Small samples are noisy — be cautious about conclusions.
• Larger samples give better estimates — so OCR questions often ask "is your estimate more or less reliable than…" with bigger n always more reliable.
• You can never get the exact theoretical probability with a finite sample, only approach it.

Quantitative example

A bag contains an unknown ratio of red and blue counters. After 50 draws (with replacement), you find 30 red.

Estimate of P(red) = 30/50 = 0.6.

After 500 draws, 285 red. P(red) ≈ 0.57. Higher confidence; the 0.6 was a slight overestimate.

After 5000 draws, 2810 red. P(red) ≈ 0.562. Likely the true probability is close to this.

OCR phrasing patterns

OCR sets these as 1–3 mark questions, typically structured:

• "Why is sample size of 1000 better than 50 for estimating?" → law of large numbers, less variance.
• "Predict the relative frequency for 10 000 trials." → it will be close to (but not exactly) the theoretical probability.

OCR mark scheme conventions

• B1 for "more reliable with larger samples" / "law of large numbers" / "less affected by random variation".
• B1 for "approaches the theoretical probability".
• "Closer to" or "approximately equal to" expected.