How did Pierre de Fermat calculate the fair division of stakes in an interrupted game?

Pierre de Fermat calculated the fair division of stakes in an interrupted game by analyzing the probabilities of all possible future outcomes.1

In his 1654 correspondence with Blaise Pascal, Fermat addressed the problem of how to divide stakes when a game is stopped before completion. He reasoned that one should define a set of equally likely future scenarios required to determine a winner. By comparing the number of outcomes that would lead to a victory for each player, he established the appropriate ratio for the distribution of the stakes.1

For example, in a game where two players are interrupted with one player needing two points to win and the other needing one, Fermat identified four equally likely sequences of future play. He determined that the player closer to victory had a 3:1 advantage, and he divided the total stakes accordingly. This approach, based on counting equally probable cases, was a foundational contribution to the early development of probability.1