How Probability Theory Got Its Start

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

- BY JACK MURTAGH Jack Murtagh

IMAGINE YOU AND I DECIDE TO PLAY 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 soon the score is 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 yet 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