The gambler's fallacy is the belief that a run of one result makes the opposite more likely. In roulette, red has the same 18-in-37 chance after five reds as after none. Random events have no memory, and a surplus is not repaid; it just becomes a smaller share of the total.
After a long run of one result, many people feel the other one is due. That feeling is common, and it is wrong whenever the events are independent.
What the fallacy says
The gambler’s fallacy is the belief that a past run changes the odds of the next outcome. After five heads, tails feels overdue. After five reds on a roulette wheel, black feels overdue. Both feelings treat the coin and the wheel as if they remembered.
Why it is wrong: independence
Two events are independent when the result of one does not change the probability of the other. A fair coin has no memory: the probability of heads is 50% on every flip, whatever came before. A roulette wheel works the same way.
On a European wheel, red covers 18 of 37 pockets, so the chance of red on a spin is 18/37, or 48.65%. After five reds in a row it is still 18/37.
Rare runs are not signs of change
Five reds in a row is unusual before it happens: is about 2.7%. But once four reds have come up, the chance that the next one is also red is not 2.7%. It is 48.65%. A run of ten reds has a probability of about 0.074%, roughly 1 in 1,350 for a given set of ten spins, and over thousands of spins such runs do occur.
The law of large numbers does not repay a surplus
A common reply is that results “even out”. What evens out is the proportion, not the surplus. In a hypothetical example, 10 flips give 8 heads, a surplus of 6 over tails. The next 990 flips are expected to give about 495 heads. The total becomes about 503 heads out of 1,000, which is 50.3%, near one half, yet the surplus of heads over tails is still about 6. The proportion converged without any correction.
The opposite mistake
Some players believe in the reverse: that a run will continue because the table or the machine is hot. It is the same error in the other direction. Independence rules out both. For slots in particular, see the guide to hot and cold slots.
What to do with this
- Treat every spin or flip as new.
- Do not raise a bet because a result looks due.
- Remember that some games, such as blackjack, do have memory in the shoe, which is why counting exists; see card counting for beginners. Roulette and slots do not.
See also: Streak lab; Myth or fact? A gambling quiz.
Frequently asked questions
Does a long streak mean the odds have changed?
No. For independent events such as roulette spins or coin flips, the odds stay the same after any streak.
Does it all even out in the long run?
The proportion of outcomes approaches the expected one. The absolute surplus is not repaid; it just becomes a smaller share of the total.
