Mathematics
Law of Large Numbers
Quick fact
The law of large numbers was first formalized by Jacob Bernoulli in his 1713 work Ars Conjectandi; it is the mathematical guarantee that insurance companies can predict losses despite unpredictable individual claims.
Why this is interesting
When you flip a coin many times, the percentage of heads gets closer and closer to 50%—but why doesn't a streak of tails guarantee a heads next time?
Read the full explanation
Understanding Law of Large Numbers
Imagine rolling a fair die. The expected average value over many rolls is 3.5, but with just a few rolls you might get an average of 2 or 5. Roll the die 10,000 times, and the average will be extremely close to 3.5. This is the law of large numbers in action: as the number of independent observations increases, the sample average converges to the true expected value. Importantly, it doesn't mean that the number of heads after 100 flips will be exactly 50—it means the proportion of heads gets arbitrarily close to 0.5 with high probability. The law explains why we can rely on long-run frequencies in gambling, science, and everyday decisions.
A deeper explanation
The law of large numbers works because the variability of the sample mean shrinks as the sample size grows. Mathematically, the standard deviation of the sample mean equals the population standard deviation divided by the square root of the sample size (σ/√n). As n increases, this standard deviation approaches zero, so the distribution of the sample mean becomes increasingly concentrated around the population mean. There are two main versions: the weak law (convergence in probability) and the strong law (almost sure convergence), both formalizing the same intuition. This principle is why you can trust opinion polls with thousands of respondents, why casinos always make money over time, and why experimental results become reliable with enough trials. It does not, however, guarantee that a short-term imbalance will be 'corrected'—that misinterpretation is the gambler's fallacy.