In the summer of 1654, Blaise Pascal laid the foundations of the laws of chance

An exceptional scientist, Blaise Pascal, in collaboration with Pierre de Fermat, laid the foundations of probability theory in the summer of 1654. A geometry of chance with far-reaching consequences for contemporary science and philosophy, explains mathematician Etienne Ghys, in his Carte blanche for “Le Monde”.

Blaise Pascal was born four hundred years ago, on June 19, 1623. Famous for his Pensées and Provinciales, he was also an exceptional scientist. I’d like to mention just one aspect of his work here: the introduction of probability, in collaboration with Pierre de Fermat, in the summer of 1654.

This theory “joins the rigorous demonstrations of science to the uncertainty of chance, and reconciling these apparently contrary things can, drawing its name from both, rightly claim this astounding title: the geometry of chance”, as he explained in Adresse à l’académie parisienne. The consequences for contemporary science, but also for philosophy, are considerable: chance has its own laws, which we understand better and better.

The starting point consists of two seemingly innocuous questions put to Pascal by a socialite, the Chevalier de Méré. The first is easy enough. If you throw two dice, it’s clearly more likely that you won’t get a double six than that you will. On the other hand, if we allow ourselves to repeat the experiment several times, we understand that after a certain number of throws, it becomes more probable to obtain a double six at least once than to obtain none at all. The question is to determine this number. The problem could be asked today at the baccalaureate, and the answer is 25.

The second question is much more subtle and was solved in a fascinating exchange of letters between Pascal and Fermat. Two people play heads or tails, and it’s agreed that whoever wins three games first will receive, say, 100 pistoles. After three games, the first player has won twice and the second once. Due to an unforeseen event, the game has to be interrupted. It’s clear that the first player has the advantage, but the second could still catch up.

“The spirit of geometry

How do you divide up the 100 pistols so that nobody feels cheated? I’ll leave it to Le Monde readers to ponder this problem. Pascal’s and Fermat’s approaches are distinct and complementary. Of course, they don’t stop there and discuss situations in which the number of players and the number of games are arbitrary.

In truth, neither of them discusses probabilities in the true sense of the word, and this problem is often considered to be as much the source of probability theory as of decision theory, in which agents must make “rational” choices. We recognize Pascal’s “heart and reason”, and the famous “spirit of finesse” alongside the “spirit of geometry”.

Some time after this exchange of letters, on November 23, 1654, “from about half past ten in the evening until about half past midnight”, Pascal experienced his second conversion and his “night of fire”. He turned his back on science to devote himself entirely to religion. He wrote to Fermat: “For, to speak frankly to you about geometry, I find it the highest exercise of the mind: but at the same time I know it to be so useless that I make little difference between a man who is only a geometer and a skilled craftsman.”

Pascal did, however, find a surprising use for geometry in 1657. While suffering from an excruciating toothache, he sought to occupy his mind with a question that would divert his thoughts from the pain. He pondered a geometry problem that dated back at least to Galileo, concerning a curve that was then called a “roulette wheel”. He thought about it so intensely that he not only found a solution, but also laid the foundations for integral calculus. The next day, the toothache had disappeared. I wouldn’t dare advise my reader to prefer the geometer’s roulette wheel to the dentist’s!