Written by Dackohn; last reviewed November 2026.
The law of large numbers says that as a sample grows, its average gets closer to the true average. With 10 coin tosses, the share of heads falls between 45% and 55% only 25% of the time. With 1,000 tosses, it does so 99.9% of the time. It says nothing about the next toss.
The numbers
How often does the share of heads in fair coin tosses land between 45% and 55%?
| Tosses | Chance of 45–55% heads | SD of the share of heads |
|---|---|---|
| 10 | 24.6% | 15.8 points |
| 100 | 72.9% | 5.0 points |
| 1,000 | 99.9% | 1.6 points |
| 10,000 | over 99.99% | 0.5 points |
The spread shrinks with √n, so a hundred times the data gives one-tenth the spread.
The gambler’s fallacy
After ten tails in a row, is heads “due”? No. The chance of heads is still 50%, because the coin has no memory. The law does not work by balancing past results. Early imbalances simply get diluted by what follows.
In fact, the gap in counts tends to grow, while the gap in percentage shrinks. After 1,000 tosses, the number of heads is typically about 16 away from 500, which is 1.6%. After 10,000 tosses, it is typically about 50 away from 5,000, which is only 0.5%. The count gap is larger, and the share is closer to 50%.
Why casinos and insurers rely on it
A casino can lose on one spin, but with a small expected edge per bet and millions of bets, its total share becomes highly predictable. Insurers do the same with claims. The law protects whoever plays the most rounds with the edge, which is not the individual gambler.
Law of large numbers vs Central Limit Theorem
| Law of large numbers | Central Limit Theorem | |
|---|---|---|
| Says | The sample mean gets close to the true mean | The sample mean’s distribution becomes normal |
| Tells you | Where the mean ends up | The shape of its fluctuation |
You can watch both in the Central Limit Theorem simulator.
When it fails
- Biased samples: a huge sample from the wrong population converges to the wrong answer.
- Dependent data: if observations are strongly linked, the effective sample size is much smaller.
- Heavy tails: for a Cauchy distribution, the mean is undefined and averages never settle. The simulator’s heavy-tailed option shows this.
The “law of small numbers”
People often expect small samples to look like the population, so they read too much into a few results. A hospital with few births will often see 60% boys on a given day, and a large one rarely does. Small samples vary more, which is exactly what the table above shows.
Try it yourself
Central Limit Theorem Simulator →Watch sample means settle down as n grows. Binomial Calculator →Exact probabilities for coin tosses.Frequently asked questions
What is the law of large numbers in simple terms?
The bigger the sample, the closer its average tends to be to the true average.
What is the difference between the law of large numbers and the gambler's fallacy?
The law says long-run averages settle down by dilution. The fallacy wrongly claims that short-run imbalances must be corrected by the next outcomes.
Does the law of large numbers apply to everything?
It needs a defined mean and data that are not strongly dependent. Heavy-tailed distributions such as the Cauchy do not obey it.
Related reading: Expected value explained · The empirical rule · Regression to the mean
Sources & further reading: NIST/SEMATECH e-Handbook of Statistical Methods · OpenStax Introductory Statistics. How we compute & validate: methodology. Written & reviewed by Dackohn; last reviewed August 2026.