How did Blaise Pascal use expectation to solve the interrupted-game problem?

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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 1654, Blaise Pascal and Pierre de Fermat addressed the "problem of points," which concerns the fair division of stakes when a game of chance is interrupted. While Fermat calculated the solution using probabilities, Pascal introduced the concept of "expectation" to solve it.1

Pascal’s method relied on evaluating the potential outcomes of the remaining rounds. Using the example of a game where player A has two points and player B has one (with each having wagered 32 pistoles), Pascal reasoned as follows:

If player B were to win the next round, the score would be tied at two points each, entitling each player to 32 pistoles.
Pascal treated this potential outcome as a fair game for the 32-pistole stake, giving each player an "expectation" of 16 pistoles.
By adding this expectation to the 32 pistoles player A was already guaranteed, Pascal determined that player A should receive 48 pistoles, while player B should receive 16.1

This approach helped formalize the theory of chances and remains a foundational concept in probability.1