New York gambling-house operator, Fat the Butch, who made millions booking dice games, was challenged in 1952 by a bigtime gambler known as The Brain to a simple wager. The bet was an even-money proposition that the Butch could not throw a double-six in 21 rolls of the dice. On the face of it, the edge seems to be with Butch. After all, there are 36 possible combinations that could come up when throwing two dice, from 1-1, 1-2, 1-3, to 6-4, 6-5, 6-6. Intuition would suggest, therefore, that 18 throws should give you a 50-50 chance of throwing any one of these combinations, including a double-six. In 21 throws, the chance of a double-six should, therefore, be more than 50-50. On this basis, the Butch accepted the even bet at \$1,000 a roll. After twelve hours of rolling, the Brain was \$49,000 up, at which point the Butch called it a day, sensing that something was wrong with his strategy.

The Brain had in fact profited from a classic probability puzzle known as the Chevaliers Dice problem, which can be traced to the 17th French gambler and bon vivant, Antoine Gombaud, better known as the Chevalier de Mere.

The Chevalier would agree even money odds that in four rolls of a single die he would get at least one six. His logic seemed impeccable. The Chevalier reasoned that since the chance that a 6 will come up in any one roll of the die is 1 in 6, then the chance of getting a 6 in four rolls is 4/6, or 2/3, which is a good bet at even money.

If the probability was a half, he would break even at even money. For example, in 300 games, at 1 French franc a game, he would stake 300 francs and expect to win 150 times, returning him 150 francs for each win with his stake returned on each occasion (total of 300 francs). With a probability of 2/3, he would expect to win 200 times, yielding a good profit.

It is easy to show intuitively that this reasoning is faulty, for if it were correct, then we would calculate the chance of a 6 in five rolls of the die as 5/6. And that therefore the chance of a 6 in six rolls of the die would be 6/6 = 100%, and in 7 rolls, 7/6!!! Something is therefore clearly wrong here.

Still, even though his reasoning was faulty, he continued to make a profit by playing the game at even money. To see why, we need to calculate the true probability of getting a 6 in four rolls of the die. The key idea here is that the number that comes up on each roll is independent of any other rolls, i.e. dice have no memory. Since each event is independent, we can (according to the laws of probability) multiply the probabilities.

So the probability of a 6 followed by a 6, followed by a 6, followed by a 6, is: 1/6 x 1/6 x 1/6 x 1/6 = 1/1296.

So what is the chance of getting at least one six in four rolls of the die?

Since the probability of getting a 6 in any one roll of the die = 1/6, the probability of NOT getting a 6 in any one roll of the die = 5/6.

So the chance of NOT getting a 6 in four rolls of the die is:

5/6 x 5/6 x 5/6 x 5/6 = 625/1296

So the chance of getting at least one 6 is 1 minus this, i.e. 1 (625/1296) = 671/1296 = 0.5177, which > 0.5.

So, the odds are still in favour of the Chevalier, since he is agreeing even money odds on an event with a probability of 51.77%.

This was all very well as long as it lasted, but eventually the Chevalier decided to branch out and invent a new, slightly modified game. In the new game, he asked for even money odds that a pair of dice, when rolled 24 times, will come up with a double-6 at least once. His reasoning was the same as before, and quite similar to the reasoning employed by the Butch.

If the chance of a 6 on one roll of the die is 1/6, then the chance of a double-6 when two dice are thrown = 1/6 x 1/6 (as they are independent events) = 1/36.

So, reasoned the Chevalier, the chance of at least one double-6 in 24 throws is: 24/36 = 2/3.

So this is very profitable game for the Chevalier. Or is it?

No it isnt, and this time Monsieur Gombaud paid for his faulty reasoning. He started losing. In desperation, he consulted the mathematician and philosopher, Blaise Pascal.

Pascal derived the correct probabilities as follows:

The probability of a double-6 in one throw of a pair of dice = 1/6 x 1/6 = 1/36.

So the probability of NO double-6 in one throw of a pair of dice = 35/36.

So, the probability of no double-6 in 24 throws of a pair of dice = 35/36 x 35/36   24 times = 35/36 to the power of 24, i.e. (35/36)24  = 0.5086.

So probability of at least one double-6 is 1 minus this, i.e. 1 0.5086 = 0.4914, i.e. less than 0.5

Under the terms of the new game, the Chevalier was betting at even money on a game which he lost more often than he won.

It was an error that Fat the Butch was to repeat almost 300 years later!

Meantime, that letter from the Chevalier de Mere to Blaise Pascal was to lead to a historic correspondence between Pascal and Pierre Fermat (of Fermats Last Theorem fame) which was to lay the groundwork of modern probability theory. All from a dice game!