Flip a coin once, and you have no idea whether it'll land heads or tails. Flip it a million times, and you can bet with near-certainty that close to 500,000 will be heads. That gap — between total unpredictability on a single trial and near-perfect predictability over many trials — is what the Law of Large Numbers describes.

Here's the precise statement: as the number of independent trials (repetitions of the same random process, where the outcome of one doesn't affect the outcome of the next) grows larger, the sample mean — the actual average of the outcomes you observed — gets closer and closer to the expected value — the theoretical long-run average predicted by probability. This getting-closer process is called convergence. The Law of Large Numbers says the sample mean converges to the expected value as the number of trials goes to infinity.

Notice what this claim does not say. It doesn't say the sample mean will exactly equal the expected value after some fixed number of trials. It doesn't say deviations get corrected or "paid back." It says that as you keep piling on trials, the average gets arbitrarily close to the true expected value, and stays close, with the ratio of how far off you are to how many trials you've run shrinking toward zero.

Example. Flip a fair coin, where heads = 1 and tails = 0. The expected value per flip is $0.5$ (since heads and tails are equally likely). Suppose in your first 10 flips you get 7 heads. In your first 10,000 flips you get 5,050 heads. Solution. After 10 flips, your sample mean is $7/10=0.70$ — a full 0.20 away from the expected value of 0.5. That's a huge relative deviation. After 10,000 flips, your sample mean is $\mathrm{5,050}/\mathrm{10,000}=0.505$ — only 0.005 away from 0.5. The absolute number of "excess" heads actually grew (from 2 extra heads to 50 extra heads), but the average moved much closer to 0.5. That's convergence: not that errors vanish, but that they shrink relative to the growing number of trials.