On the evening of 18 August 1913, at a roulette table in the Casino de Monte-Carlo, the ball fell on black. Then it fell on black again. And again. Somewhere past the tenth black, word rippled through the gilded salon and a crowd began to gather; somewhere past the fifteenth, the crowd began to bet. Their reasoning felt like iron: the wheel must balance out, red was overdue, and every fresh black only made the next red more certain. So they doubled and redoubled their stakes on red — and the wheel, which knew nothing of what it owed anyone, delivered black twenty-six times in a row. By the time the streak broke, the casino had reportedly raked in millions of francs. It should be said that the tale rests on casino legend and later retellings rather than sworn records, and details vary with the teller — but the event became the founding parable of probability’s most expensive mistake, which is why the gambler’s fallacy is also known as the Monte Carlo fallacy.
The fallacy is the belief that independent random events keep score — that a run of one outcome makes the opposite outcome more likely, because chance is somehow obliged to even things up. It is false for a simple, brutal reason: the wheel has no memory. Each spin of a European roulette wheel offers black a fraction under half — 18 of the 37 pockets — and that probability is identical whether black has just appeared once or twenty-five times. What fooled the crowd was a confusion between two different questions. Before any spins, the chance of twenty-six blacks in a row is genuinely fantastical — ignoring the green zero, about one in 67 million, and longer still with the zero counted. But after twenty-five blacks have already happened, the improbability is spent. The twenty-sixth spin is just a spin.
The deeper misunderstanding concerns the law of large numbers. It is true that over millions of spins the proportions of red and black drift towards their true probabilities — but chance achieves this by swamping, not by compensating. Early imbalances are never corrected; they are simply drowned in an ocean of later spins. Psychologists Amos Tversky and Daniel Kahneman diagnosed the human side in the 1970s: we expect even short random sequences to look “representative” of fairness, a bias they wryly called belief in the law of small numbers. The same wiring shows up far from the casino — in lottery players who avoid last week’s numbers, in parents certain that three sons make a daughter due, and in studies suggesting that even judges and loan officers grow reluctant to rule the same way too many times in a row.
The house that probability built
Monte Carlo was purpose-built to profit from such wiring. The casino was developed in the 1860s under François Blanc, the entrepreneur who ran it for the tiny principality of Monaco and helped make the Grimaldis’ statelet rich enough to spare its citizens income tax. Blanc’s trademark was the single-zero wheel: thirty-seven pockets, numbers 0 to 36, giving the house its slim, relentless edge of about 2.7 per cent. No streak, system or martingale can beat it, because no arrangement of bets changes the arithmetic of independent spins — doubling after every loss merely guarantees that when the bad run comes, it finds you betting your maximum.
That has never stopped the legends. In 1891 Charles Wells, a serial fraudster from Britain, won hundreds of thousands of francs over several visits and “broke the bank” repeatedly — meaning a table’s cash reserve was exhausted and play paused while it was replenished. Fred Gilbert’s music-hall song of 1892, “The Man Who Broke the Bank at Monte Carlo”, made him immortal; the courts, in time, made him a convict, and he died poor. Wells claimed an infallible system. The wheel, as ever, had other plans.
The neatest twist in the story is that Monte Carlo’s name was eventually redeemed by science. In the 1940s, at Los Alamos, Stanisław Ulam — convalescing and playing solitaire — realised that hard physics problems could be attacked by running thousands of random simulated trials and counting outcomes. He developed the idea with John von Neumann for work on nuclear weapons, and their colleague Nicholas Metropolis christened the secret technique the “Monte Carlo method”, a nod to the casino where Ulam’s uncle used to borrow money to gamble. Today Monte Carlo simulation prices financial derivatives, forecasts weather and elections, and tests the safety of reactors — randomness, properly respected, turned into an instrument of precision.
That is the real moral of the night of 18 August 1913. Randomness is not malicious, and it is not just; it is merely indifferent, and it cannot be owed. The players at that table lost fortunes to a phantom debt that red never promised to pay. The wheel has no memory — only the players do.
Quiz nuggets
- On 18 August 1913, black reportedly came up 26 times in a row at the Casino de Monte-Carlo, costing gamblers who kept backing red millions of francs — the event that named the Monte Carlo fallacy.
- Ignoring the green zero, the odds of 26 same-colour spins in a row are roughly 1 in 67 million; a European wheel has 37 pockets and a house edge of about 2.7 per cent.
- Charles Wells “broke the bank” at Monte Carlo in 1891, inspiring Fred Gilbert’s 1892 song “The Man Who Broke the Bank at Monte Carlo”.
- Tversky and Kahneman called the psychology behind the gambler’s fallacy “belief in the law of small numbers” (1971).
- The Monte Carlo simulation method was devised in the 1940s at Los Alamos by Stanisław Ulam and John von Neumann, and named by Nicholas Metropolis after the casino.