Law of large numbers

The law of large numbers states a fundamental fact about randomness: when the same random experiment is repeated a very large number of times, the observed frequency of a result approaches its theoretical probability. In other words, the actual behavior of a large sample eventually reflects, faithfully, the chances calculated on paper.

The mental image is that of an average settling down. Over a few trials, the results can look chaotic and lopsided; five heads in a row is nothing unusual. But as the repetitions accumulate, the occasional deviations cancel out and the overall proportion begins to swing more and more weakly around the expected value.

In practice, one tracks the proportion of a result, that is the number of occurrences divided by the number of trials. Over 10 flips of a fair coin, getting 7 heads gives a proportion of 0.7, far from 0.5. Over 10,000 flips, the total number of heads will typically be very close to 5,000, and the proportion will be near 0.5.

It is important to understand what the law actually claims: it is the relative frequency that converges, not the gap in absolute numbers. The exact count may keep drifting away from a perfect half of heads, while the proportion itself tightens around the probability.

The classic trap is the gambler's fallacy. The law never says that an outcome that is overdue should catch up in order to rebalance the series. When the draws are independent, the past does not influence the next trial: the convergence comes from the sheer mass of new trials, not from a correction of the old ones.

This principle sheds light on how the site's tools are used. A number generator, a repeated die roll, or a coin flip may seem unfair over a short run, but their fairness reveals itself over the long term, as the frequencies catch up with the stated probabilities.