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How did Blaise Pascal’s arithmetic triangle help solve gambling problems?

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French philosopher Blaise Pascal — Blaise Pascal Blaise Pascal, oil painting by an anonymous artist, c. 1650; in the Musée Carnavalet–Histoire de Paris, Paris. Pascal laid the foundation for probability theory.

In his 1665 posthumous work, Blaise Pascal demonstrated that his arithmetic triangle—an array now commonly known as Pascal’s triangle—provided a systematic method for solving elementary gambling problems.1

The triangle functions by arranging numbers such that each interior entry is the sum of the two entries directly above it. These entries correspond to binomial coefficients, which represent the number of combinations possible when selecting a subset of objects from a larger set.23

By using the triangle to calculate these combinations, Pascal was able to group them to determine the odds and fair distribution of stakes in games of chance. This work was central to his correspondence with Pierre de Fermat, which established the foundations of the modern theory of probability. While the array is named after Pascal, it is important to note that it was known to Islamic and Chinese mathematicians, such as Al-Karajī and Jia Xian, centuries earlier.1423