06 / 10 · Probability
The law of large numbers
Why luck evens out
Flip a fair coin 10 times and you might get 7 heads. That is 70%, far from the 50% you expected. Nothing is wrong: small samples are noisy.
Keep flipping. After 100 flips you will usually be within about 10% of a half. After 10,000, within about 1%. The running fraction of heads settles down towards 0.5. This is the law of large numbers.
fraction of heads = heads ÷ flips typical distance from 0.5 ≈ 0.5 ÷ √flips 100 flips → ±0.05 10000 flips → ±0.005
Try it · Flip the coin in batches and watch the line. Press New run a few times: every run wanders at first, then they all squeeze towards 0.5.
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Careful: the coin has no memory. After five heads in a row, the next flip is still 50/50. The fraction evens out not because tails catch up, but because the early streak is drowned out by thousands of later flips.
This law is what makes casinos and insurance companies reliable: one bet is a gamble, a million bets are close to certain.
Key ideas
Words worth remembering
- Relative frequency
- How often something happened, divided by how many tries.
- Law of large numbers
- With more tries, the relative frequency gets closer to the true probability.
- Gambler's fallacy
- The false belief that past results change the odds of independent events.
Think about it
Why does the √ from the square roots sections show up in how fast luck evens out?