De Méré’s paradox and the birth of probability theory

Posted: July 2, 2018 in history of science, probability
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Taking a break from studying thermodynamics: 17th century soldiers playing at dice

In 17th century France, dice games were a popular and fashionable habit. All kinds of people played at dice – soldiers, sailors, socialites, aristocrats, celebrities … and professional gamblers. Every era in history has featured this latter group, often colourful characters with sharp minds living off their wits.

Such an individual was Antoine Gombaud (1607-1684), who had adopted the high-flown title of Chevalier de Méré and was known as a flambuoyant big spender in gambling circles. He also had good connections with Renaissance intellectuals, and was himself a notable Salon theorist.

Professional gamblers have one overriding aim in life, which is to win. This is easier said than done however as gambling necessarily involves games of chance, and chance is a fickle creature. So the professional gambler has to proceed with caution until he finds a wager whose odds are in his favor. Then he brings big money to the table, plays long and hard, and walks away a richer man.

Calculating success

This was how Gombaud operated. But unlike most other gamblers, he did not rely solely on experience to show him which bets were favorable. He had an analytical turn of mind, and had started to work out on the basis of mathematical principle whether a certain game had favorable odds.

He first applied his thinking to a popular dice game in which players wagered on a six appearing in four throws of a die. He correctly reasoned that since each number on a six-sided die was equally likely to occur, the chance of getting a six on a single throw must be 1/6. He then considered the chance of getting a six if a die were thrown four times instead of once. He reasoned that the chance of success would be four times greater since each throw represented a separate opportunity for a six to occur, and he calculated that chance as 4 x 1/6 = 2/3. In other words, the odds of winning were favorable (i.e. >1/2).

Long before the law of large numbers was formulated, Gombaud seems to have intuitively understood that these favorable odds meant that although the outcome of an individual game could not be predicted, success would be assured if enough games were played.

Gombaud made piles of money out of this game, thereby cementing belief in his method of mathematical analysis as a means of identifying a winning bet. Buoyed by this success he extended the same reasoning to another game where he calculated that the odds of winning were favorable. But an unpleasant surprise was in store.

Unexpected losses

Gombaud’s new focus of attention was a dice game in which players wagered on getting a double six in twenty four throws of two dice. He correctly reasoned that the chance of getting a double six in a single throw of two dice was 1/36. Then applying his formula he calculated the chance of success as 24 x 1/36 = 2/3, the same favorable odds as in the previous game.

Emboldened by this analysis, Gombaud brought a stack of money to the dice table to wager on getting double sixes. But his expectations did not materialise; in fact the more he played this game, the more his losses mounted. Gombaud simply could not understand it. Both games had exactly the same favorable odds. So why did he win at one and lose at the other?

Desperate for an explanation of his losses, he wrote in 1654 to one the foremost thinkers of his time, Blaise Pascal (1623-1662), who in turn shared the news of “De Méré’s paradox” with the profoundly talented amateur mathematician Pierre Fermat (1607-1665). The two began a legendary correspondence, out of which the theory of probability was born and the paradox was solved.

Fallacious reasoning

The work of Pascal and Fermat revealed Gombaud’s mistake in thinking that the chances of success with n throws could be calculated by multiplying the chance of success for a single throw by n.

Taking the first game as an example, Gombaud thought that after two throws of the die the chance of success doubled from 1/6 to 1/3. But a chart of all 36 possible outcomes shows that the total number of favorable outcomes (shown in gold) is not 12, but 11.

Pascal and Fermat no doubt saw that the ratio of favorable outcomes to total outcomes after two throws could be written as

and after three throws as

and so on. Since 5/6 was the chance of failure in a single throw, and the exponent was the number of throws, the formula for the chance of getting at least one success in n throws could be generalised as

where q is the chance of failure in a single throw. This was the correct formula for computing whether the odds were favorable or not. Contrast this with the formula Gombaud used

It is instructive to compare Gombaud’s incorrect formula with the correct one for the first game

and the second game

The correct figures in the final column make it clear why Gombaud won at the first game and lost at the second. They also show what a knife-edge situation it was. If the second game had been played with just one more throw (n=25, 1-q^n = 0.506) Gombaud would have won both games. There would have been no paradox to explain, and the genius minds of Pascal and Fermat might never have been applied to founding probability theory!

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Probability profiles

Game 1

Game 2

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P Mander November 2017