My June diary included the following brainteaser.
Antoine Gombaud was a 17th-century French gent, and a gambler. Knowing that the chance of getting a six on one roll of a die is one-sixth, he figured that the chance of getting a six on four rolls is four times one-sixth, which is two-thirds. Betting even money that he'd get a six in four rolls would therefore be a nice little earner.
So it proved. Gombaud's math was actually wrong; but it was wrong in the right direction, so to speak, and he came out ahead.
Thus encouraged, he cooked up a new game. The chance of getting a pair of sixes on a roll of two dice is, he figured (correctly) one-thirty-sixth. On twenty-four rolls of a dice pair, therefore, the chance of getting at least one pair of sixes should be twenty-four times one-thirty-sixth — two-thirds again!
Gombaud's math was wrong again; but this time it was wrong in the wrong direction. Betting evens on this new game, he couldn't stay ahead.
He grumbled about this in 1654 to Blaise Pascal, a real first-rank mathematician. Pascal kicked it around in some correspondence with Pierre de Fermat, another of the same. Out of their correspondence emerged foundations for the modern Theory of Probability.
A nice bit of mathematical folklore. But where did Gombaud go wrong in his math?