April 19, 2026

7 min read

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How a Renaissance gambling dispute spawned probability theory

A dispute over how to divvy up the pot in an interrupted game of chance led early mathematicians to invent modern risk assessment

By Jack Murtagh edited by Jeanna Bryner

Amanda Montañez

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Imagine you and I are playing a simple game of chance. We each throw $50 into a pot and start flipping a coin. Heads, you get a point; tails, I get one. The first person to reach 10 points walks away with the full $100. The game gets underway, and the score is currently eight to six in your favor. Suddenly my phone rings: there’s an emergency, and I must leave in a hurry. Now we have a problem. You don’t want to just hand me my $50 back because you’re winning. But I’m reluctant to give you the whole pot because I still have a chance to hit a lucky streak and mount a comeback. What is the fairest way to split the cash?

Known as the “problem of points,” or “problem of the division of the stakes,” this puzzle stumped mathematicians for more than 150 years. And it did so for good reason: probability theory hadn’t been invented when the problem was first posed. Two greats of 17th-century math, Blaise Pascal and Pierre de Fermat, corresponded about the problem in a famous series of letters. They not only discovered the correct way to share the pot but also created the foundations of modern probability theory in the process. To this day, the solution is the basis for risk assessments of all kinds, helping us make smarter bets on everything from buying a stock to insuring a home along a coastline.

In 1494, Italian mathematician Luca Pacioli first took an early crack at the problem of points in his textbook, the title of which translates to Summary of Arithmetic, Geometry, Proportions and Proportionality. He proposed that players should split the pot in proportion to how many points they each have at the time of interruption. In our running example, you have won eight of the 14 flips thus far. According to Pacioli’s solution, you would take eight fourteenths of the pot, which equals about $57.14. I would take the remaining six fourteenths. The solution sounds sensible, but more than 50 years later, Niccolò Fontana “Tartaglia” noticed that it failed in cases where the point ratio between players was extreme. What if the interruption came after a single coin toss? Under Pacioli’s rule, the winner of that one flip would take the entire pot, even though the game was far from decided. This would be clearly unfair—and the problem of points is all about seeking a fair split.

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Tartaglia proposed an alternative method. Imagine that, in our hypothetical game, you’re ahead by two flips. You have one fifth of the 10 flips needed to win. Because that’s one fifth closer to the goal, Tartaglia reasoned that you should get your full stake back and take one fifth of my stake: the original $50 you put in plus one fifth of my $50, for a total of $60. This new approach seems to operate more equitably, especially at the extremes. Now if the game got interrupted after one flip, then the winner of that flip would take only one tenth of their opponent’s stake instead of all of it. While Pacioli’s method rewards the winning player based on the size of their lead relative to the number of flips thus far, Tartaglia’s method rewards them based on the size of their lead relative to the total length of the game. Tartaglia doubted his own innovation, though, writing, “In whatever way the division is made there will be cause for litigation.” He believed that no perfect mathematical solution existed and that the problem was designed to cause arguments. It turns out he was at least right to doubt his own solution. Imagine that one player had 199 points and that the other had 190 points during a game with a goal of 200 points. Tartaglia would award the first player only nine two-hundredths of their opponent’s stake, or $2.25, even though their opponent would need 10 tails in a row to win. The first player’s measly payout hardly seems to reflect their overwhelming likelihood of winning at that stage of the game.

The debate