Empirical samples and the Law of Large Numbers
The bigger your sample, the closer the relative frequency gets to the theoretical probability. This is the Law of Large Numbers (LLN) in plain English.
What does the LLN say?
Imagine flipping a fair coin many times. After 10 flips you might see 7 heads (0.7), or 3 (0.3) — wide variability. After 1,000 flips, you'd expect to see ≈ 500 heads, with the relative frequency hovering near 0.5. After 1,000,000 flips, the relative frequency is astonishingly close to 0.5.
Formally: as the number of trials n → ∞, the relative frequency of an outcome converges to its theoretical probability.
Implication for the GCSE: large samples are more reliable
If you want an estimate of an unknown probability:
- 10 trials: rough.
- 100 trials: better.
- 1,000+ trials: much better.
The size of the random fluctuation roughly scales as 1/√n. So to halve your error, you need 4 times the trials. To get 10× the precision, 100× the trials.
Estimation in practice
Suppose a quality-control inspector tests N items and finds f are defective. Estimate P(defective):
P(defective) ≈ f / N
The bigger N, the better the estimate. Typically you'd be told the sample size and asked to comment on reliability.
Predicting outcomes from estimates
Once you have a good estimate of PA, you can predict counts in further trials.
Worked example: in 500 spins of a spinner, blue came up 215 times. Estimate P(blue), and predict the number of blues in 1500 further spins.
- P(blue) ≈ 215/500 = 0.43.
- Expected blues in 1500 = 1500 × 0.43 = 645.
Comparing samples — which estimate to trust?
Two students each collect data:
- A: 20 trials, 9 successes (P̂ = 0.45).
- B: 200 trials, 80 successes (P̂ = 0.40).
Whose estimate is more reliable? B, because of the larger sample. The variability of A's estimate (±1/√20 ≈ ±0.22 rough scale) is much greater than B's (±1/√200 ≈ ±0.07).
You don't need to write the rough error formulas — just say "B has more trials so the estimate has less random variation."
Combining samples
Two samples for the same experiment can be combined by adding successes and trials.
Example: A finds 9/20, B finds 80/200. Combined: 89/220 = 0.4045.
This is better than either individual estimate — but only if the experiments really are testing the same thing under the same conditions.
⚠Common mistakes— Common mistakes (examiner traps)
- Trusting a small sample as much as a large one.
- Thinking probabilities "balance out". After 10 heads in a row, the next toss is still 0.5 (the Gambler's Fallacy).
- Comparing samples by absolute count, not proportion. 9/20 (0.45) is bigger than 80/200 (0.40), even though 80 > 9.
- Using the LLN to argue for a particular result on the next trial. It's about long-run averages, not predicting individual outcomes.
- Reading "reliable" as "close to 0.5". Reliability depends on sample size, not on the value itself.
➜Try this— Quick check
A coin is tossed 30 times and lands heads 18 times. A friend says, "The next 30 tosses must have more tails to balance out." Is this correct? Why?
No — the Gambler's Fallacy. Each toss is independent; previous outcomes don't affect future ones. The Law of Large Numbers describes long-run convergence, not short-term compensation.
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