Law of Large Numbers

With just a few observations, results can vary widely by chance. With many observations, random variations tend to cancel out, and the average becomes more predictable.

For example, if you toss a fair coin ten times, you might get seven heads, a 70% rate. If you toss it 10,000 times, the proportion of heads will almost certainly be very close to 50%.

The law of large numbers is the foundation of insurance. An insurer cannot predict whether any individual home will catch fire, but across hundreds of thousands of policies, it can predict fairly accurately how many claims it will receive. This allows it to set premiums that cover expected claims and costs. The same principle helps lenders estimate default rates across large portfolios of loans and casinos rely on it to make steady profits.

The law of large numbers does not mean that results balance out in the short run. Believing that a run of losses makes a win more likely is a mistake known as the gambler's fallacy. Each independent event, such as a coin toss, has the same probability regardless of what happened before.