How did Blaise Pascal help lay the foundation of probability theory?

Blaise Pascal helped establish the modern theory of probabilities primarily through a 1654 correspondence with the mathematician Pierre de Fermat. Their collaboration was sparked by a problem posed by the chevalier de Méré regarding the fair division of stakes in an interrupted game of chance.12

While Fermat approached the problem using the concept of chances, Pascal introduced the influential mathematical concept of "expectation". To solve such problems, Pascal utilized the "arithmetic triangle"—now commonly known as Pascal’s triangle—which allowed for the calculation of combinations. His work, detailed in a 1665 posthumous publication, demonstrated how to group these combinations to resolve elementary gambling dilemmas. This systematic use of calculation to address problems of chance was a hallmark of the 17th-century scientific revolution, helping to transition probability from a subject of mere speculation to a rigorous mathematical discipline.23