The Law of Large Numbers

Flip a fair coin a few times and you might get three heads in a row, or four tails. Early on, the running proportion of heads bounces around wildly because each flip carries a lot of weight when there are only a handful of flips total. Flip it 1000 times and one weird streak barely matters anymore. Each new flip adds a smaller and smaller fraction to the total, so the running proportion wobbles less and less and drifts toward 0.5. That settling isn't magic or correction, it's just dilution: more data drowns out the early noise.

The mechanism is purely about sample size, not about the coin trying to balance itself out. The coin has no memory. After five tails in a row, the probability of heads on the next flip is still 0.5, because each flip is an independent trial. Nothing about the past outcomes changes the physical process generating the next one. The gambler's fallacy is believing the coin is 'due' for heads. It isn't due for anything; it just keeps producing 0.5 chances, flip after flip, and long-run averages take care of the rest.

This idea underpins why larger samples give trustworthy estimates: a big sample's proportion is less affected by random swings than a small sample's, so it lands closer to the true population proportion or probability. That connects directly to sampling distributions later in the unit, where the spread of a statistic shrinks as sample size grows. The law of large numbers is about long-run behavior, not about any single outcome or short run, which is exactly where the gambler's fallacy goes wrong.