I recently read a book that reviews the basic notions of probability. And this book also describes the "Gambling of the Chevalier De Mere": Solving this problem, Blaise Pascal produced the basic laws of probability.

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.

But Pascal found that the Chevalier’s assessment was wrong. Here is the correct solution:

• First Game: The probability of not getting a six when rolling a die 4 times is P1 = (5/6)⁴ . This means (P(one) + P(two) + P( three) + P(four) + P(five))⁴ . And the wanted result is the probability to get a six when rolling a die 4 times is 1- P1 = 1 - (5/6)⁴= 0.518
• Second game: The probability of not getting a six when rolling 2 dice 24 times is P2 = (35/36)²⁴ . So the probability to get a six when rolling 2 dice 24 times is 1 - P2 = 1 - (35/36)²⁴ = 0.492

This explains why the Chevalier was winning at the first game but losing at the second one.

Gambling and the Chevalier De Mere