On an evening in the summer of 1913, at a casino in Monte Carlo, a wheel came up black. Then black again. It kept coming up black, and by the time the run ended it had done so twenty-six times in a row.
What happened around that table is the reason the evening is still written about. As the run went on, more and more people bet on red, and they bet more and more heavily, because red was surely due. It had been so long. It could not stay black.
It could. The wheel has no memory. On every single spin the chance of black was the same as it had been on the first, and the twenty-five spins already behind it changed nothing about the twenty-sixth. A very long run is rare — but once you are standing inside one, it is no less likely to continue than it ever was.
The mistake was so clearly displayed that evening that it took the name of the town. It is called the Monte Carlo fallacy, or simply the gambler's fallacy, and it has nothing to do with gambling in particular. It is the belief that independent events somehow keep score.
A coin that has landed heads five times has a precisely even chance on the sixth throw. So does a coin that has landed tails five times. So does a coin nobody has thrown yet.