Betting gave rise to probability theory, most notably through the famous "problem of points" (today we would say division of the stakes): two players bet on random but "equally likely" outcomes, as we would now say, scoring one point for each win. They stake the same amount (12 ducats each). The first to reach a certain number of points—say, 10—wins the entire pot. But then the game is interrupted before it ends, with the score at 8–4. How should they divide the stakes?
One problem, three solutions, one theory
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Luca Pacioli was the first to discuss this problem, in 1494. His solution was to divide the stakes in proportion to the points scored, giving 16 ducats to the first player and eight to the second.
More than 50 years later, Tartaglia refuted Pacioli's reasoning, though without providing an acceptable solution. Cardano proposed dividing the stakes in proportion to the number of points the other player still needed. Thus, in the previous example, the player in the lead would leave with 18 ducats and the other with six.
Another century passed before Pascal solved the problem in his correspondence with Fermat, while studying a question posed by the Chevalier de Méré. This marked the birth of probability theory, as well as the concept of expected value, now expressed not in ducats but in pistoles. Incidentally, could you divide the 24 pistoles? (See the answer at the end of the article.)
