“It’s all just the law of averages.” People say this after a long losing run, expecting a win to arrive soon, or after a long winning run, expecting it to end. The phrase sounds like mathematics, but it is not a mathematical law at all. It is a popular misreading of a real result, and the gap between the two explains a lot of confusion about chance.
What the real law says
The genuine result is the law of large numbers. Roughly, it says that as the number of independent trials grows, the observed proportion of an outcome tends to get closer to its true probability. Flip a fair coin many times and the share of heads will typically sit near one half.
Notice what it does not say. It does not say the count of heads and tails will balance. It does not say that early imbalance gets cancelled. It does not say anything about what happens on the next flip, which is still a fresh trial with the same chances.
The “law of averages” version
The informal version goes: “I have had lots of tails, so heads must come soon to even things out.” This treats the coin as if it keeps a running tally and corrects it. It does not. A fair coin, a die, or a roulette wheel has no memory of what came before. This belief is the gambler’s fallacy, and the law of averages is the story people tell themselves to justify it.
An illustration with numbers
These numbers are illustrative, chosen so you can check the arithmetic. Imagine you plan 100 flips of a fair coin, and the first 10 produce 8 heads and 2 tails.
- Expected heads in the remaining 90 flips: 90 × 1/2 = 45. The coin does not try to “catch up” by producing extra tails.
- Expected total heads after 100 flips: 8 + 45 = 53, which is 53 per cent.
- Now imagine 1,000 flips instead. Expected total: 8 + 495 = 503 heads, which is 50.3 per cent.
The early surplus of 3 heads over the “fair share” (8 against an even split of 5 in the first ten, so a surplus of 3) is not removed. It is simply diluted. In percentage terms it fades as the total grows: 53 per cent, then 50.3 per cent. The proportion converges towards one half even though the gap in raw numbers is never corrected.
Why the raw gap can actually grow
There is a further surprise. The typical size of the gap between heads and tails does not shrink with more flips. For a fair coin, the standard deviation of the number of heads in n flips is half the square root of n. For 100 flips that is 5 heads; for 10,000 flips it is 50 heads. So the typical wobble around the expected count grows with more trials, but it grows more slowly than the number of trials, so as a proportion it shrinks: 5 in 100 is 5 per cent, while 50 in 10,000 is 0.5 per cent.
This is why talk of results “evening out” is misleading. Proportions settle; counts do not need to.
Why short runs do not even out
Short run results are dominated by luck, and no mechanism rebalances them. A hundred spins can sit well away from the long-run average and nothing about that is unusual or “owed” to anyone. With small sample sizes, runs and lopsided results are normal, which is why a short record, whether of a player, a coin or a game, says very little about underlying odds.
The same misunderstanding shows up in the opposite direction. After a winning run, some players assume things are about to turn, which is just the same error with the sign flipped. Nothing in the next outcome depends on the last.
Where the idea feels right
Two real effects help the myth survive:
- Regression to the mean. An unusually good or bad result is often followed by one nearer to average, simply because extreme results are rare. This is not a correction mechanism; it reflects that unusual results were mostly luck to begin with.
- Averages across many players. If many people play, the overall proportions across the group are very stable, even though no individual’s luck balances out. Large totals look smooth, and that smoothness gets mistaken for individual fairness.
What to take from this
If you want to judge a game, look at its structure: the probabilities and the payouts, which determine the house edge and the long-run result. Do not judge it by what “should” happen next. Over time, casino games produce a loss on average, and the law of large numbers is exactly what makes that average reliable for the operator and unreliable as comfort for any individual player’s short session.
Gambling carries financial risk and can be harmful. If it is affecting your life, a local support service can help.