# The Law of Large Numbers, Explained With Coin Flips Numbers · 6 min read · updated 2026-10-29 > 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 depend on it. ## The short answer The law of large numbers says that as you repeat a random process more and more times, the average of the results settles toward the true average. A fair coin may give 70% heads in ten flips but will almost always be within a point or two of 50% after ten thousand. It describes proportions settling, not outcomes being corrected, and the reverse is just as important: small samples swing wildly. Flip a coin ten times and you might easily get seven heads: 70%. Flip it ten thousand times and you’ll almost certainly end up between 49% and 51%. That settling-down is the law of large numbers, and it’s the reason casinos stay in business, insurance companies set prices and opinion polls work at all. It’s also one of the most misunderstood ideas in probability, and the misunderstandings cost people money. So let’s be careful about what it says and what it doesn’t. ## What it says The law of large numbers says: as you repeat a random process more and more times, the *average* of your results gets closer and closer to the true underlying average. For a fair coin the true share of heads is 50%. After 10 flips the share can be anywhere. After 100 it’s usually between 40% and 60%. After 10,000 it’s very likely between 49% and 51%. After a million it’s within about a tenth of a point. Use the toy at the top to watch the line wobble wildly at first and then settle down. The idea was first proved by Jacob Bernoulli, who spent about twenty years on it and called it his “golden theorem”. It was published in 1713, after his death, in a book called *Ars Conjectandi*, the art of conjecturing. He’d seen what we still see: with enough repeats, chance becomes predictable. ## A man, a coin and a prison camp In 1940, a South African mathematician named John Kerrich was visiting Denmark when the Germans invaded. He was interned for the duration of the war, and with plenty of time on his hands, he did an experiment. He flipped a coin 10,000 times and recorded every result. He ended with 5,067 heads, or 50.67%: within about two thirds of a point of 50%. The interesting thing is a second number. The *gap* between heads and tails, the raw count, was larger at the end than at the start. At 100 flips the two were within a dozen of each other. At 10,000 they were 134 apart. The proportion had settled, but the absolute difference had grown, as it will for a fair coin. *(An interactive illustration belongs here. Try it on the page: https://cognosc.com/blog/law-of-large-numbers)* It’s not specific to coins. Roll a die and the average roll will settle toward 3.5, even though you can never roll a 3.5. Measure the height of people in a town and the average of a thousand will be close to the average of everyone. The shape of the process doesn’t matter much. What matters is that each result is an independent draw from the same source, and that you take plenty of them. ## 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* earlier results. Suppose you start with ten heads in a row. The coin won’t produce extra tails to make up for them; that’s [the gambler’s fallacy](/blog/gamblers-fallacy). What happens is that after 10,000 more flips, those first ten heads are swamped. They’re still ten extra heads, but ten out of ten thousand is a rounding error. The proportion settles; the raw difference wanders. The early luck is diluted, never repaid. It also doesn’t say that a *short* run will balance. The law is silent about ten flips. It speaks about ten thousand. Treating a handful of results as though they must reflect the long-run average is what Kahneman and Tversky called the “belief in the law of small numbers”. And it doesn’t fix **bias**. The law guarantees that a large, fair sample gets close to the truth. If your sample isn’t fair, if it systematically leaves out the people you care about, more data makes you more confidently wrong. In 1936, the magazine *The Literary Digest* polled more than two million people and predicted Alf Landon would beat Franklin Roosevelt in a landslide. Roosevelt won in a landslide, carrying 46 of the 48 states. The sample was huge and badly chosen, drawn from lists of car owners and telephone subscribers in the middle of the Depression. ## Why small samples mislead The flip side of the law is that small samples swing a lot, purely by chance. It sounds obvious. It catches almost everyone. - **Small hospitals** have more days when over 60% of babies born are boys, because with few births, lopsided days are common. A big hospital’s share of boys sits close to half every day. - **A new player** who made 7 of their first 10 shots isn’t necessarily better than one who made 5. Ten shots is a very small sample. - **The best and the worst** performers in almost any ranking tend to be the small ones, as the next story shows. ## The most dangerous equation In the 2000s, the Gates Foundation and other funders put hundreds of millions of dollars into breaking up large schools into smaller ones. Their reasoning: when researchers looked at the top-performing schools in the US, a surprising number were small. The statistician Howard Wainer pointed out a problem. If you looked at the *worst*-performing schools, a surprising number were also small. Small schools aren’t better or worse on average. They’re just more *variable*, because their averages come from fewer pupils, so a handful of excellent or struggling students moves the whole figure. Wainer called the underlying equation, which says that the spread of an average shrinks with the square root of the number of observations, “the most dangerous equation”, because ignoring it has led to so many expensive mistakes. The same pattern shows up in health. Researchers who mapped rates of kidney cancer across US counties noticed that the lowest rates were in sparsely populated, rural counties. Then they noticed that the *highest* rates were too. Small populations mean noisy rates, and it’s the noise that creates the extremes, not anything in the water. *(An interactive illustration belongs here. Try it on the page: https://cognosc.com/blog/law-of-large-numbers)* ## Why businesses depend on it - **Casinos** have a small edge on every bet: 2.7% on a single-zero roulette wheel, around 5.3% on a double-zero one. Any one gambler can win big. Across millions of bets, the casino’s share settles at its edge with near certainty. The law of large numbers turns a small edge into a reliable profit. - **Insurers** can’t predict whether *your* house will burn down. They can predict, quite accurately, how many of a hundred thousand houses will. The more customers they have, the steadier the claims, which is why big insurers can price more tightly than small ones. - **Pollsters** ask a thousand people to learn about millions. The margin of error shrinks as the sample grows, though not very quickly: 1,000 people give about ±3 points, and you’d need 4,000 to halve it to ±1.5. - **Scientists** run the same measurement many times, because the average of many measurements is much more precise than any single one. ## One exception worth knowing The law has conditions. It needs each repeat to be a fair draw from the same process, and the process needs to have a stable average. Some distributions don’t. The Cauchy distribution, which describes certain kinds of extreme, fat-tailed randomness, has no average at all, and the average of a million Cauchy draws is no more settled than the average of one. Nature is mostly better behaved. But where extreme outliers dominate, such as financial crashes, a reassuring average can be much less reassuring than it looks. ## What to take from it Two things, one on each side of the coin. When you see an average from a large, fair sample, trust it: that’s the law working for you. When you see an extreme result from a small one, be careful: that’s the law working against you, because small samples throw up extremes all the time. And remember the thing the law *doesn’t* say. Luck isn’t repaid. It’s only outnumbered. To see where your own instincts land, the free [probability test](/test/probability) has several questions on sample size, and the [statistics test](/test/statistics) covers polls and averages. ## What to remember - More repeats means the average settles near the true value, and the range of likely error shrinks like one over the square root of the sample. - Proportions settle, but raw differences don’t shrink: early luck is diluted, never repaid. - Small samples are extreme more often than large ones, which is why both the best and worst results come from small groups. - Casinos and insurers make money from large numbers; individual gamblers and customers live in the swing. - A big sample won’t fix a biased one: the law only works if each repeat is a fair draw. ## Questions people ask ### What does the law of large numbers say in simple terms? If you repeat a random experiment many times, the average of what you get gets closer and closer to the true underlying average. Flip a fair coin often enough and the share of heads will settle near 50%. ### Does the law of large numbers mean results even out? Only in the sense of proportions. A coin that starts with 10 extra heads doesn’t make up for them later: after thousands more flips the early excess is a small part of the total, so the share is near 50%, but the extra heads are still there. That’s dilution, not correction. ### What’s the difference between the law of large numbers and the gambler’s fallacy? The law says averages settle over many repeats. The gambler’s fallacy wrongly says that a streak must be balanced out by an opposite streak soon. The first is true because early results get swamped. The second is false because each event is independent. ### Why do casinos always win in the long run? Each bet carries a small built-in edge, such as 2.7% on a single-zero roulette wheel. Any one gambler can win, but across millions of bets the casino’s share of the money settles at that edge with near certainty. The law of large numbers turns a small edge into a reliable profit. ### Does a bigger sample always give a better answer? A bigger sample reduces random error, but not bias. A poll of two million people who all share some feature will confidently give the wrong answer. The law of large numbers works only when each observation is a fair, independent draw. ## Sources and further reading - Bernoulli, J. (1713). Ars Conjectandi. Basel: Thurneysen Brothers. - Tversky, A. and Kahneman, D. (1971). Belief in the law of small numbers. Psychological Bulletin, 76(2), 105–110. - Wainer, H. (2007). The most dangerous equation. American Scientist, 95(3), 249–256. - Freedman, D., Pisani, R. and Purves, R. (2007). Statistics (4th ed.). W. W. Norton. (Kerrich’s coin-tossing experiment.) - Gelman, A. and Nolan, D. (2002). Teaching Statistics: A Bag of Tricks. Oxford University Press. --- Canonical page: https://cognosc.com/blog/law-of-large-numbers From Cognosc (https://cognosc.com): free tests, articles and an AI tutor that has you reason things through.