The Gambler's Fallacy and Other Probability Traps

After a long run of reds, it feels as if black must be due. It isn't. The wheel has no memory, and twenty million simulated spins show exactly that. But probability intuition fails in both directions, and one famous "fallacy" turned out to be partly wrong itself.

Monte Carlo, 1913

The best-known example happened at the Monte Carlo Casino in August 1913, when the ball fell on black 26 times in a row. As the streak grew, players bet more and more on red, convinced that the run had to end. Most of them lost heavily.

How unlikely was it? On a single-zero wheel the chance of any 26-spin run of one colour, starting from a given spin, is about 1 in 67 million. That is very rare for any particular sequence of 26 spins. But casinos around the world spin wheels millions of times a year, so long streaks happen somewhere all the time.

Testing it: 20 million spins

We simulated 20,000,000 spins of a European wheel and recorded what happened on the spin immediately after each run of reds.

The theoretical probability of red is 18/37 = 48.65%. After three, five or eight reds, the next spin is red with the same probability, within simulation noise. Nothing is "due".

Why the intuition is so strong

People expect short sequences to look like long-run averages, a tendency psychologists call belief in the "law of small numbers". The real law of large numbers says that proportions settle down over many trials. It does not say that past deviations get corrected. An early excess of reds is not balanced by extra blacks later. It is diluted by the huge number of spins that follow.

The opposite error

The mirror image is the "hot hand": believing that a run of wins makes further wins more likely. For roulette wheels and dice that is just as wrong. For human performance, the story is more interesting.

In 1985, Gilovich, Vallone and Tversky analysed basketball shooting and concluded that the hot hand was an illusion. In 2018, economists Joshua Miller and Adam Sanjurjo showed that the original analysis contained a subtle statistical bias. In a finite sequence, the observed rate of hits right after a streak of hits is expected to be lower than the overall rate, even with no hot hand at all. Once they corrected for that bias, the same data showed evidence of streak shooting.

The lesson is not that streaks are real in roulette. It is that probability intuition is hard even for experts. The only protection is to write the problem down, or simulate it.

Where the fallacy does not apply

The gambler's fallacy applies only to independent events. When draws are made without replacement, as in a blackjack shoe, the past does change the future. That is exactly why card counting works and roulette systems don't. We tested the most common roulette systems in a separate simulation.