# The Paradox Behind Modern Probability Theory

### How gambling and bad reasoning set the foundations of modern probability theory.

Dice games were popular in France in the XVII century.

People from all social classes played these games, from soldiers to celebrities to French aristocrats.

Some played professionally, like Antoine Gombaud, a writer and philosopher who, despite not being a nobleman, called himself “Chevalier De Méré,” or “Knight from the Méré region.”

The self-styled Chevalier was obsessed with gambling and the mathematics behind each game and spent his days looking for an edge to keep his winnings flowing.

The Chevalier loved playing a simple game where players bet on getting a six when rolling a die four times.

De Méré figured the chance of getting a six when rolling a die was 1 in 6. Every new roll of the dice gave him another opportunity to get a six, so his chances of winning were four times better or 2 out of 3. With these favorable odds, De Méré kept winning against the other unsuspecting players.

But a variation of this game changed everything for the Chevalier. This time, players bet on getting a double six when rolling two dice twenty-four times.

At first, De Méré concluded that getting a pair of sixes on a single roll of two dice had the same odds as rolling two sixes on two rolls of one die, or 1 out of 36. Therefore, throwing the dice twenty-four times would be 24 x 1/36 = 2/3. This looked identical to the previous game!

Armed with this information, the Chevalier kept betting, but, to his surprise, he started losing.

Desperate and frustrated because he couldn’t figure out why his analysis wasn’t helping, De Méré enlisted two mathematicians to help him with the mystery: Blaise Pascal and Pierre de Fermat took up the challenge. After several letters, they solved the riddle and laid out the foundations for the modern probability theory.

Although De Méré thought both games had the same winning odds, they didn’t.

For the first game, his chances of not getting a six in one roll of a die were 5 out of 6. If the Chevalier rolled four times, his chances of not getting a six were (5/6)⁴. Therefore, his chances of getting at least one six were 1 - (5/6)⁴ or 52%.

The second game was different. De Méré chances of not getting a double six in one roll of a die were 35 out of 36. If he rolled twenty-four times, his chances of not getting a double six were (35/36)²⁴. Therefore, Méré chances of getting at least one double six were 1 - (35/36)²⁴, or 49%.

Losing money on the second game wasn’t about bad luck but a mistake in the Chevalier’s reasoning. With 49% odds, he didn’t stand a chance of keeping his winning streak.

This problem is known today as **De Méré’s Paradox**, a veridical paradox where, counterintuitively, the results look absurd but is demonstrated to be true nevertheless.

De Méré’s Paradox teaches us how deceiving the most obvious things can be and how important it is to question our assumptions.

Very interesting story Santiago!

It makes you think twice before assuming something!

I like reading this type of stories; keeps me close.