# Gambling and the Chevalier De Mere

Archeologists have discovered dice dating back several thousand years B.C. Modern dice games, however, only grew popular in the middle ages. They contributed to entertainment with or without exchange of material goods involved.

The chances of winning in such a game were very much assessed with a blend of wishful thinking, and some sort of logical reasoning - sometimes correctly and, at other times, incorrectly.

Historically recorded is the fait of a swinging Flemish renaissance gentleman, the Chevalier de Mere. Around 1650, he suffered severe financial losses for assessing incorrectly his chances of winning in certain games of dice. Contrary to the ordinary gambler, he pursued the cause of his error with the help of Blaise Pascal. He reached fame because in the process the area of probability was created. - Let us take a look at what happened.

Among other things, the Chevalier systematically tried his luck with the following two games.

• First game:   Roll a single die 4 times and bet on getting a six.
He assessed his chances of winning as follows. The chance of getting a 6 in a single throw is 1 out of 6. Therefore, the chance of getting a 6 in 4 rolls is 4 times 1 out of 6; i.e. 2 out of 3.
• Second game:   Roll two dice 24 times and bet on getting a double six.
He assessed his chances of winning as follows. The chance of getting a double six in one roll is 1 out of 36. Therefore, the chances of getting a double six in 24 rolls is 24 times 1 out of 36; i.e. 2 out of 3.

To his painful surprise the Chevalier ended up loosing badly with the second gamble. He was desparate for an explanation, and so he sought help from one of the great thinkers of his time, Blaise Pascal (1623-1662). After a careful analysis, Pascal was able to spot the Chevalier's error.

In the process he discovered a fundamental principle for assessing the probability for a certain event, amongst a collection of possible events, to occur. This fundamental principle is just as valid now as it was then. It is broadly used and constitutes the germinating point for the theory of probability.

Correct analysis of the Chevalier's dice games   Let us now follow in Pascal's footsteps and analyze correctly the chances of winning in these two games.

Single die   Rolling a single die once leads to precisely one of 6 possible outcomes: Exactly one of the numbers 1,2,3,4,5,6 will be on top. The die is called fair, if each of these outcomes is equally likely. Players of dice games usually assume that the dice they are using are fair. So let us assume this too:

If we roll a die 4 times, then the total number of all possible outcomes is

6 x 6 x 6 x 6 = 1296

Out of these there are

5 x 5 x 5 x 5 = 625

outcomes with no 6 in them.

Thus, if we bet on getting at least one 6 when rolling a die 4 times, there are

• 625   possibilities of losing, and
• 1296 - 625 = 671   possibilities of winning

This means that our chances of winning with this game are higher than our chances of losing.

Two dice   Let us now turn to the two-dice game. Rolling two dice once leads to one of 36 possible outcomes, namely all possible outcomes of rolling die number 1 combined with all possible outcomes of rolling die number two. Thus, if we roll two dice 24 times, then the total number of possible outcomes is

36 x 36 x ... x 36    (36 multiplied with itself 24 times)

which is approximately    22,452,257,707,350,000,000,000,000,000,000,000,000.

Out of these there are

35 x 35 x ... x 35    (35 multiplied with itself 24 times)

which is approximately    11,419,131,242,070,000,000,000,000,000,000,000,000   outcomes with no double 6.

Thus, if we gamble on getting at least one double 6 when rolling two dice 24 times, there are approximately

• 11,419,131,242,070,000,000,000,000,000,000,000,000 possibilities of losing, and
• 22,452,257,707,350,000,000,000,000,000,000,000,000 - 11,419,131,242,070,000,000,000,000,000,000,000,000 = 11,033,126,465,280,000,000,000,000,000,000,000,000 possibilities of winning

This means that the chances of winning with this game are lower than the chances of loosing - as the Chevalier De Mere learnt the hard way.