At 11, he was solving scientific mysteries. Built one of history's earliest calculators. But a gambling problem led him to crack the mathematics of chance

Games of chance have existed for centuries, but gambling also helped mathematics develop a new way to understand uncertainty. In the 17th century, a seemingly simple question about how to divide money when a game ended early brought together two leading mathematicians, Blaise Pascal and Pierre de Fermat, and helped lay the foundation of modern probability theory.

Pascal was already an extraordinary mathematician long before gambling entered the story. At 11, he wrote an essay on the sounds produced by vibrating bodies. By 16, he had developed a theorem in geometry, and before turning 19, he had built a mechanical calculator to help his father with tax calculations.

The story is documented by APS News and the Encyclopedia of Mathematics, which describe Pascal's 1654 correspondence with Fermat over the “Problem of Points” as a major step in the development of probability. Pascal introduced mathematical expectation and used a recursive method to work out the problem.

Pascal was proving mathematical ideas as a child

Born in Clermont-Ferrand, France, in 1623, Pascal was educated largely by his father, Étienne Pascal, a mathematician and local magistrate. After the family moved to Paris, Étienne introduced his son to leading mathematicians of the period.

At 11, Pascal wrote about vibrating bodies. A year later, he produced his own proof that the angles of a triangle add up to two right angles.

By 16, he had written an essay on conic sections that established what became known as Pascal's Theorem. It states that when a hexagon is inscribed in a conic section, the three intersection points of its opposite sides lie on a straight line.

The work was so advanced that René Descartes initially doubted that the teenager had written it himself.

Before probability, he built a mechanical calculator

Pascal's mathematical ability soon found a practical purpose. His father had become responsible for tax calculations in Rouen and faced endless additions and recalculations.

Before turning 19, Pascal began constructing a mechanical calculator to help him. Known as the Pascaline, the machine could perform addition and subtraction.

The Encyclopedia of Mathematics says Pascal eventually built and sold 50 of the machines, some of which still exist. Its development showed that his mathematical work was not limited to abstract theories.

A gambling problem posed a difficult question

In 1654, French essayist and amateur mathematician Antoine Gombaud, who was interested in gambling, approached Pascal with the “Problem of Points.”

The question had first been raised in 1494 by Italian mathematician Luca Pacioli. If a game ended before either player reached the winning score, how should the money at stake be divided?

Consider a game in which six goals are needed to win. If play stops when one player has five goals and the other has three, the player in front should receive more money. But exactly how much more?

Pascal realised that the answer had to account for each player's chances of winning if the game continued.

Pascal and Fermat turned chance into mathematics

Pascal began corresponding with Pierre de Fermat about the puzzle. Their letters produced mathematical methods for calculating the fair division of the stake.

Fermat's approach involved listing possible outcomes and determining how many favoured each player. In one example, a player leading 2-1 in a best-of-five coin-toss game would be favoured by three of four possible outcomes and should receive three-fourths of the stake.

Pascal developed a different approach. He considered the possible results of each successive round and then worked backward from the final outcomes to calculate each player's chances at the point where the game had been interrupted.

The Encyclopedia of Mathematics credits Pascal with introducing mathematical expectation and using it recursively to solve the Problem of Points. This helped probability theory move beyond simply counting possible outcomes.

His work eventually went far beyond gambling

Pascal's method could not handle every kind of uncertainty. Problems such as weather or stock markets could not always be reduced to a finite list of equally likely outcomes.

Later mathematicians extended the field. By the early 18th century, Jakob Bernoulli had developed the law of large numbers, while further advances eventually made probability important in economics, actuarial science and the social sciences.

Pascal later abandoned games of chance after a near-fatal carriage accident in 1654 prompted a profound religious experience. He largely turned away from mathematics and science, although he briefly returned to the subject in 1658-59 to study the cycloid.

He died in 1662 at the age of 39. His work on a gambling puzzle had helped establish a mathematical framework for understanding something far larger than games: uncertainty itself.