Mathematics
The Law of Large Numbers and the Gambler's Fallacy
Quick fact
In a fair coin toss, the probability of heads is always 50%, regardless of previous outcomes; after a run of five heads, the next toss still has a 50% chance of heads. The law of large numbers only guarantees that the overall average will approach 50% over many tosses—it does not predict a 'correction' in the short run.
Why this is interesting
After flipping a coin and getting heads five times in a row, do you feel a tail is 'due'? That feeling is one of the most common statistical illusions—and it has a name.
Read the full explanation
Understanding The Law of Large Numbers and the Gambler's Fallacy
Imagine you flip a fair coin ten times. You might get seven heads and three tails—a noticeable imbalance. But if you flip it ten thousand times, the percentage of heads will be very close to 50%. This is the law of large numbers: as the number of trials grows, the average result gets closer and closer to the expected value. The gambler's fallacy is the mistaken belief that after a streak of one outcome, the opposite is 'due' to happen to balance things out. In reality, each flip is independent; the coin has no memory. The law works over the long run, not by compensating for short-term deviations but by drowning them out in the sheer number of trials.
A deeper explanation
The law of large numbers is a mathematical theorem that states that as the number of independent trials increases, the sample average of the outcomes converges to the expected value. For example, if you roll a fair die, the expected value is 3.5. After a few rolls, the average might be far from 3.5, but after thousands, it will be extremely close. This happens because the random fluctuations from each trial tend to cancel each other out as more trials are added. However, these fluctuations do not 'remember' the past—each roll is independent. The gambler's fallacy arises from a misunderstanding of this principle: people expect a short-term 'balancing' that does not occur. In fact, the law of large numbers assures only that the long-run relative frequency will align with the true probability. This distinction is critical in fields from gambling to medicine to finance, where misjudging randomness can lead to poor decisions.