How did Pierre de Fermat and Blaise Pascal help found probability theory?

The modern mathematics of chance is widely attributed to a 1654 correspondence between French mathematicians Pierre de Fermat and Blaise Pascal. Their collaboration was sparked by a problem posed by the gambler Antoine Gombaud, chevalier de Méré, regarding the fair division of stakes in an interrupted game of chance.1

While both mathematicians arrived at the same numerical result, they utilized different conceptual approaches:

• Fermat calculated the solution based on the probability of future outcomes, determining the odds by counting equally likely sequences of play.1
• Pascal approached the problem using the concept of expectation, calculating the value of a player's share based on the potential outcomes of the remaining rounds.1

This exchange established the foundational principles of probability theory. Pascal later contributed to the field through his work on the "arithmetic triangle," which provided a method for calculating combinations to solve gambling problems. Their work moved the study of chance into the realm of rigorous mathematical calculation, a hallmark of the 17th-century scientific revolution.1