Chevalier de Méré's Problem

A 17^th century gambler, the Chevalier de Méré, made it to history by turning to Blaise Pascal for an explanation of his unexpectd losses. Pascal combined his efforts with his friend Pierre de Fermat and the two of them laid out mathematical foundations for the theory of probability.

Gamblers in the 1717 France were used to bet on the event of getting at least one 1 (ace) in four rolls of a dice. As a more trying variation, two die were rolled 24 times with a bet on having at least one double ace. According to the reasoning of Chevalier de Méré, two aces in two rolls are 1/6 as likely as 1 ace in one roll. (Which is correct.) To compensate, de Méré thought, the two die should be rolled 6 times. And to achieve the probability of 1 ace in four rolls, the number of the rolls should be increased four fold - to 24. Thus reasoned Chevalier de Méré who expected a couple of aces to turn up in 24 double rolls with the frequency of an ace in 4 single rolls. However, he lost consistently.

Explain

Chevalier de Mere's Problem

Compare two problems:

What is the probability of having at least one 1 in four rolls of a dice?
What is the probability of having at least one double 1 in 24 rolls of two die?

Solution to Problem 1

Four rolls of a dice may have one of 6⁴ equiprobable outcomes. Of these, 5⁴ are unfavorable leaving (6⁴ - 5⁴) favorable for the bet. The probability of getting at least 1 ace is then

(6⁴ - 5⁴) / 6⁴	= (1296 - 625) / 1296
	= 671 / 1296
	≈ 0.5177 ...
	> 0.5

showing that the odds are in favor of the bettor.

Solution to Problem 2

One double roll has 36 equiprobable out comes of which 35 are unfavorable to the bet. In 24 rolls there are 36²⁴ possible outcomes of which only (36²⁴ - 35²⁴) are favorable. Thus the probability of winning the bet equals

(36²⁴ - 35²⁴) / 36²⁴	= 1 - (35/36)²⁴
	≈ 1 - 0.5086
	= 0.4914
	< 0.5.

References

J. Bewersdorff, Luck, Logic & White Lies, A K Peters, 2005
R. Falk, Understanding Probability and Statistics, A K Peters, 1993

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