Pascal and Fermat Found the Theory of Probability
A gambler's question about unfinished dice games turns into five letters that create a new branch of mathematics
Quick facts
- Correspondence
- 5 letters, summer 1654
- Core problems addressed
- The dice problem; the problem of points
- Pascal's method
- Backward induction / expected value
- Limitation
- Solved fully only for two players
What happened
In the summer of 1654, Blaise Pascal exchanged a series of five letters with Pierre de Fermat that laid the foundation for the theory of probability. The correspondence addressed two problems already studied in less general form by earlier mathematicians including Cardano, Pacioli, and Tartaglia: the dice problem, asking how many times a pair of dice must be thrown before a double six can be expected, and the problem of points, asking how to fairly divide the stakes of a game of chance that is interrupted before it finishes. Pascal used a method of backward induction to work out each player's expected winnings at each possible stopping point, while Fermat provided an explicit formula using the same triangle of binomial coefficients now called Pascal's triangle, and the two independently arrived at consistent solutions. They fully solved the problem of points for two players but, as Pascal's biography notes, did not develop mathematical methods powerful enough to solve the same problem for three or more players.
Why it matters
The Pascal-Fermat correspondence turned informal gambling intuition into a mathematical framework with defined methods, expected value chief among them, and later mathematicians built the entire modern discipline of probability theory on the groundwork laid in these five letters, a foundation that today underpins statistics, insurance, and much of modern science.
How we know
Pascal and Fermat's letters from 1654 survive and have been published and analyzed by historians of mathematics, and their content, including the specific problems addressed and the limits of what the two mathematicians could solve, is documented directly in their own correspondence.
Sources
- MacTutor History of Mathematics, University of St Andrews. Blaise Pascal · Reputable sourcemathshistory.st-andrews.ac.uk · The domain "mathshistory.st-andrews.ac.uk" is on our Reputable source registry. · Link is live and its text matches the event's key terms (Jul 2026)
- Department of Mathematics and Statistics, Utah State University. The Problem of Points · Reputable sourceusu.edu · The domain "usu.edu" is on our Reputable source registry. · Link is live and its text matches the event's key terms (Jul 2026)
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