Why does the law of large numbers not mean that a coin is more likely to land heads after many tails?

The belief that a coin is more likely to land heads after a series of tails is a common misconception known as the gambler’s fallacy or the doctrine of the maturity of the chances.1

In probability theory, coin tosses are considered independent events. This means that the outcome of any single toss is not influenced by previous results; the probability of landing heads remains exactly 1/2 for every individual flip, regardless of how many tails have occurred previously.12

The law of large numbers does not imply that past outcomes are "balanced out" by future ones through some mystical force. Instead, it states that as the number of trials increases, the proportion of heads will eventually approach the theoretical probability of 1/2. This happens not because the coin "remembers" past results, but because any initial imbalance becomes statistically insignificant as the total number of tosses grows very large. The law governs the long-term ratios of outcomes, not the specific results of individual, independent events.31