Numbers · 3 min read
The Law of Large Numbers, Explained With Coin Flips
Flip a coin enough times and heads settles near 50%. What the law of large numbers says, what it doesn't, and why casinos and insurers rely on it.
By the Cognosc team ·
Flip a coin ten times and you might easily get seven heads: 70%. Flip it ten thousand times and you’ll almost certainly get between 49% and 51%. That settling-down is the law of large numbers.
Try it
Watch the average settle
Keep flipping. The line is the share of heads so far; the band is 45% to 55%.
After 0 flips: 50.0% heads. Early on it swings wildly; with more flips it settles near 50%. The coin never “catches up”: the early luck just gets diluted.
What it says
As you repeat a random experiment more and more times, the average of the results gets closer and closer to the expected value.
For a fair coin, the expected share of heads is 50%. With a few flips, the share swings widely. With many, it settles near 50% and stays there.
What it doesn’t say
This is where people go wrong. The law is about proportions, not counts, and it doesn’t work by correcting past results.
Suppose you start with 10 heads in a row. The coin won’t produce extra tails to even things out; see the gambler’s fallacy. Instead, after 10,000 more flips, those first 10 heads are swamped. The share of heads is back near 50%, even though the gap in the count, heads minus tails, is likely to be as big or bigger than when you started.
The proportion settles; the raw difference wanders. The early luck is diluted, never repaid.
Why small samples mislead
The flip side of the law is that small samples swing a lot, purely by chance. This catches people out constantly:
- Small hospitals have more days where over 60% of babies born are boys, because with few births, lopsided days are common.
- The best and worst schools in rankings are often small ones: a few strong or weak pupils move a small school’s average a long way.
- A new player who made 7 of their first 10 shots may be no better than one who made 5.
Kahneman and Tversky called the mistaken belief that small samples should look like large ones the “law of small numbers”.
Why businesses depend on it
- Casinos have a small edge on every bet. Any one gambler can win big, but over millions of bets, the casino’s share settles at its edge with near certainty.
- Insurers can’t know who will crash their car this year, but across a million drivers, the number of crashes is very predictable.
- Polls work because the average opinion in a large random sample settles close to the average in the whole population.
How fast does it settle?
Slower than you might hope. The typical wobble in a proportion shrinks with the square root of the number of tries. So to halve the wobble, you need four times as many flips. That’s why a poll of 1,000 people has a margin of error of about 3 points, and getting it down to 1.5 points needs about 4,000.
Try it
Flip the coin above: 10 times, then 100, then 10,000. Watch the line swing early, then flatten towards 50%.
The free probability test has the small-hospital question and nine more.