Key takeaway: The law of large numbers states that as the number of independent trials grows, the observed proportion or sample mean gets closer and closer to the true probability or population mean.
Grade 12 – AP StatisticsAP Statistics Unit 2Probability and Counting
Definition of Law of Large Numbers
The law of large numbers explains why casinos profit and why polls improve with size. A short run of trials can stray far from the expected result, but the average over a very long run settles near the theoretical value.
The law is about proportions and averages, not about counts. The raw gap between heads and tails can actually grow, while the share of heads drifts toward one half. It also has no memory: past results never push future trials back toward balance.
Formula and notation
For independent trials with mean \(\mu\), the sample mean \(\,\overline{x}_n\) after \(n\) trials satisfies
\[ \,\overline{x}_n\to\mu\quad\text{as } n\to\infty. \]
For a proportion, \(\hat{p}_n\to p\), where \(p\) is the true probability of success on each trial.
Key facts
- Convergence applies to the average or proportion, not to the number of heads minus the number of tails.
- Small samples vary a lot; large samples vary far less.
- Independent trials do not compensate for earlier streaks, so there is no correction coming.
- Simulations rely on this law: more repetitions give estimates closer to the true probability.
Law of Large Numbers example
In a simulation of a fair coin, 10 flips gave 7 heads, and 1,000 flips gave 507 heads. Compare the proportions and the raw differences from the expected number of heads.
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After 10 flips: \(\hat{p}=\frac{7}{10}=0.70\), and the count is \(7-5=2\) above the expected 5.
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After 1,000 flips: \(\hat{p}=\frac{507}{1000}=0.507\), and the count is \(507-500=7\) above the expected 500.
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The count gap grew from 2 to 7, but the proportion moved from 0.70 to 0.507.
The proportion of heads moved much closer to 0.5, which is what the law predicts, even though the raw count gap grew.
Common mistake
Watch out
After five tails in a row, a student argues that heads is “due” on the next flip because of the law of large numbers.
Why it is wrong
Fair-coin flips are independent, so the next flip is still 50% heads. The law works by dilution, not correction: a long future run of ordinary flips outweighs the early streak. No flip owes the past anything.
Related terms and lessons
Browse the glossary: A–Z list · AP Statistics vocabulary · Probability and Counting