One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal to the first explicit reasoning about what today is known as an expected value.
Luca Pacioli considered such a problem in his 1494 textbook Summa de arithmetica, geometrica, proportioni et proportionalità.
[1] In the mid-16th century Niccolò Tartaglia noticed that Pacioli's method leads to counterintuitive results if the game is interrupted when only one round has been played.
In that case, Pacioli's rule would award the entire pot to the winner of that single round, though a one-round lead early in a long game is far from decisive.
Tartaglia constructed a method that avoids that particular problem by basing the division on the ratio between the size of the lead and the length of the game.
[2] The problem arose again around 1654 when Chevalier de Méré posed it to Blaise Pascal.
Through this discussion, Pascal and Fermat not only provided a convincing, self-consistent solution to this problem, but also developed concepts that are still fundamental to probability theory.
Second, he showed how to calculate the correct division more efficiently than Fermat's tabular method, which becomes completely impractical (without modern computers) if
The imagined extra round may lead to one of two possible futures with different fair divisions of the stakes, but since the two players have even chances of winning the next round, they should split the difference between the two future divisions evenly.
[4] It is easier to convince oneself that this principle is fair than it is for Fermat's table of possible futures, which are doubly hypothetical because one must imagine that the game sometimes continues after having been won.
Pascal's analysis here is one of the earliest examples of using expected values instead of odds when reasoning about probability.
The direct application of Pascal's step-by-step rule is significantly quicker than Fermat's method when many rounds remain.
Through clever manipulation of identities involving what is today known as Pascal's triangle (including several of the first explicit proofs by induction) Pascal finally showed that in a game where player a needs r points to win and player b needs s points to win, the correct division of the stakes between player a (left side) and b (right side) is (using modern notation): The problem of dividing the stakes became a major motivating example for Pascal in his Treatise on the arithmetic triangle.
[4] [5] Though Pascal's derivation of this result was independent of Fermat's tabular method, it is clear that it also describes exactly the counting of different outcomes of