On 18 August 1913, at the casino in Monte Carlo, the ball on a roulette wheel landed on black again and again. As the streak grew, gamblers rushed to bet on red, convinced it was "due". Black came up 26 times in a row. The losses that night became the most famous illustration of a thinking error now known as the gambler's fallacy, or the Monte Carlo fallacy.
The fallacy in one sentence
The gambler's fallacy is the belief that after a run of one outcome, the opposite outcome becomes more likely, when each event is in fact independent.
A roulette wheel has no memory. After 25 blacks, the probability of black on the next spin of a single-zero wheel is still 18 in 37, exactly as it was on the first spin.
How unlikely was the streak?
For a single-zero wheel, the probability that a sequence of 26 spins all lands on the same colour, red or black, is:
2 × (18/37)^26 ≈ 1 in 68.4 million
That sounds impossible, but casinos spin wheels millions of times a year across the world. Over enough spins, very rare streaks are certain to happen somewhere. What is impossible is to predict when one will end.
Key point: Rare streaks and independent events are not in conflict. The streak as a whole is rare; each individual spin within it is completely ordinary.
Why our intuition fails
People expect short sequences to look "balanced", as if a small sample should mirror the long-run average. Psychologists call this the representativeness heuristic: we judge how likely a sequence is by how random it looks. Five reds in a row looks wrong, so we feel black must be coming. The wheel does not know or care.
The fallacy in casino games
| Game | How the fallacy appears | Reality |
|---|---|---|
| Roulette | "Red has come up five times, bet black" | Each spin is independent |
| Slots | "This machine hasn't paid for ages, it's due" | Each spin is independent; see the RNG |
| Baccarat | "Follow the scoreboard patterns" | Hands are almost independent |
| Lotteries | "These numbers haven't come up in years" | Every draw is independent |
A warning about streaks you see
Casinos often display recent results next to roulette wheels. The information is accurate, but it is useless for prediction, and it invites exactly the reasoning that failed in 1913.
Worked example: A worked check. After four reds on a single-zero wheel, what is the chance of red on the fifth spin? Still 18 in 37, about 48.6%. The chance of five reds in a row before any spin was (18/37)^5 ≈ 2.7%, but four of those reds have already happened, so they no longer count against the fifth.
The related mistake: chasing
The gambler's fallacy often feeds a dangerous habit: raising stakes to win back losses because a win "must" come. Systems built on that idea, such as the martingale, are covered in Why the Martingale System Fails.
Watch out: If you notice yourself betting more because a result feels "due", that is a signal to stop, not to bet. Support is available on our Responsible Gambling page.
Test the numbers yourself with the roulette odds and house edge calculator.
- In 1913, black came up 26 times in a row at Monte Carlo; the chance of 26 of the same colour is about 1 in 68.4 million.
- Each spin remains independent: after any streak, black is still 18 in 37 on a single-zero wheel.
- Rare streaks are inevitable over millions of spins; predicting when they end is impossible.
Sources
- Gambler's fallacy — Wikipedia: the Monte Carlo example
- Representativeness heuristic — Wikipedia
18+ only. No betting system changes the house edge. See our Responsible Gambling page.
RTP Data explains the mathematics of casino games, with every figure sourced or worked out in the text.