People have been gambling since before recorded history. Astragali — the knucklebones of sheep and goats — have been found at archaeological sites going back 5,000 years, and they were almost certainly used as dice. The Romans gambled. The Greeks gambled. Medieval people gambled. For thousands of years, humans threw bones and dice and drew lots without developing any systematic mathematics of chance. The question of why it took so long is more interesting than it might appear.
The usual answer is that antiquity lacked the concept of probability because it lacked the concept of randomness as a natural phenomenon. When a die fell, it fell where the gods directed. Chance was not a lawlike process to be mathematized; it was divine will to be accepted or propitiated. This is partly true, but it obscures a more technical obstacle: the calculation of chances requires thinking about the relative frequency of outcomes over many trials, and this requires holding in mind an ensemble of possibilities rather than focusing on the single actual outcome. It requires a particular kind of counterfactual reasoning that is not natural and not simple.
The breakthrough came in the 16th century, and it came from an unlikely source: Gerolamo Cardano, a Milanese physician, astrologer, and compulsive gambler who was also one of the most gifted mathematicians of his age. Cardano wrote a manuscript, never published in his lifetime, called Liber de Ludo Aleae — the Book on Games of Chance — in which he worked out, for the first time, systematic rules for computing odds. He identified that the probability of an event is the ratio of favorable outcomes to total equally likely outcomes. He understood that the probability of a sequence of independent events is the product of their individual probabilities. He knew that a fair game required equal expected value for both players. These ideas feel obvious now, but no one before him had stated them clearly.
The formal founding of probability theory is usually dated to 1654, to a correspondence between Blaise Pascal and Pierre de Fermat prompted by a question from the Chevalier de Méré — a French nobleman and experienced gambler who had encountered a problem that his intuitions could not resolve. The problem of points: if a game of chance is interrupted partway through, how should the stakes be divided between the two players? This sounds like an accounting question, but it requires computing the probability that each player would have won had the game continued, which requires reasoning about a space of possible futures. Pascal and Fermat's correspondence developed the combinatorial methods that made this calculation tractable.
Why did it take the Renaissance, specifically, for this to happen? Several things converged. Cardano, Pascal, and Fermat were working in an intellectual environment that increasingly valued calculation and quantification — the same environment that produced double-entry bookkeeping, compound interest tables, and navigational mathematics. There was a commercial motivation: merchants needed to price risk, and early insurance markets were emerging. There was also a philosophical shift: the idea that the natural world follows regular mathematical laws, rather than expressing divine caprice, was gaining ground. If the world is lawlike, then chance events might be lawlike too — not individually determined but statistically regular in the aggregate.
The next major development came from Christian Huygens, who in 1657 published the first printed text on probability, De Ratiociniis in Ludo Aleae, formalizing the concept of mathematical expectation. Then came Jacob Bernoulli, whose posthumous Ars Conjectandi (1713) proved the first version of the law of large numbers: as the number of trials increases, the observed frequency of an event converges to its probability. This was not just a technical result but a philosophical one. It said that probability was not merely a feature of our ignorance but something that constrained what the world would actually produce. Bernoulli called it the Golden Theorem.
What strikes me about this history is the contingency of the delay. The mathematics of probability is not deep in the way that Cantor's diagonal argument is deep, or the way Gödel's incompleteness theorems are deep. The basic combinatorial ideas are accessible to anyone with patience and facility with fractions. Cardano could have lived in Athens and worked out the same results; the tools were there. What was missing was not the mathematics but the questions — specifically, the questions posed by gamblers who cared enough about winning to formalize their reasoning. The mathematics of chance waited for people who needed it, and the people who needed it took their time arriving. Sometimes the obstacle between us and an idea is not intellectual but motivational: we have not yet asked the question clearly enough to need the answer.