Chance before probability
People have played with dice and knucklebones for thousands of years, and ancient and medieval games clearly involved luck. Yet there is no good evidence that anyone before the Renaissance had a general mathematical theory of chance. Outcomes were attributed to fortune, fate or the gods, and fair division of stakes was a matter of custom.
What changed was a sustained effort to put numbers on gaming questions. The history below sticks to the well-documented milestones. Where details are uncertain, they are left out.
Cardano and an early manuscript
Gerolamo Cardano, an Italian physician and mathematician who lived from 1501 to 1576, wrote a manual on games of chance, usually known by the Latin title Liber de Ludo Aleae (“Book on Games of Chance”). It was written in the 1500s but not published until 1663, after his death. It contains early reasoning about counting the favourable cases among all the possible ones for dice, which is the basic idea behind the ratio definition of probability. It also mixes in practical advice for gamblers of the day, and it was not a finished theory.
Galileo and three dice
Galileo is also credited with a short note on a question about throwing three dice: why a total of 10 turns up more often than a total of 9 even though both can be made in six combinations of values. The resolution is that you must count ordered outcomes, as the later idea of counting principles makes explicit. Each combination counts a different number of times depending on how many of its dice are equal. This is a good illustration of how counting mistakes, not ignorance of luck, troubled early thinkers.
The problem of points and the 1654 letters
The event most often cited as the birth of probability theory is a correspondence in 1654 between Blaise Pascal and Pierre de Fermat, two French mathematicians. A gambler and writer known as the Chevalier de Mere had put several puzzles to Pascal. One was the problem of points: if a game of chance between two players is interrupted before it ends, how should the stakes be divided fairly? The better-known dice question from the same circle is covered separately under the de Mere problem.
Pascal and Fermat reached the same answers by different methods. The key insight was to divide the stakes according to each player’s chances of winning from the position reached, rather than by points won so far.
Illustration (simplified, assuming each round is a fair 50/50): two players play first to 3 rounds. The game stops when player A has 2 and player B has 1. A needs one more round and B needs two. Play at most two more rounds:
- B wins only by taking both: 1/2 × 1/2 = 1/4.
- A wins in every other case: 1 − 1/4 = 3/4.
A fair split of the stakes is therefore 3 parts to A and 1 part to B. This is exactly the kind of reasoning from future possibilities that produced the idea of expected value.
Huygens spreads the ideas
Christiaan Huygens, a Dutch scientist, heard of the Pascal and Fermat exchange and wrote a short treatise on reasoning in games of chance, published in 1657. It was widely read and became a standard introduction for decades. It defined the value of a chance in terms of what it was worth on average, and worked out a number of problems with dice.
Bernoulli and the long run
Jacob Bernoulli worked for years on a book that was published after his death, in 1713, called Ars Conjectandi. Among other things it contains the first proof of what is now called the law of large numbers: with enough repetitions, observed frequencies settle close to the underlying probability. That result connects games to measurement and to real-world data, and it explains why casino results are predictable in aggregate even though single outcomes are not.
Later developments
From the 1700s onwards, probability grew beyond games. Mathematicians such as de Moivre, who studied the shape that binomial results take for many trials and anticipated the normal distribution, and Laplace, who published a major treatise in the early 1800s, applied it to astronomy, insurance and population statistics. In the twentieth century, Andrey Kolmogorov set out a rigorous axiomatic foundation in 1933, and modern probability rests on it.
Why the history matters to players
Three lessons from this story carry over to modern games.
- Counting carefully matters. Several early disputes were about whether to count ordered or unordered outcomes.
- Value comes from the future, not the past. The problem of points showed that fair shares depend on what could still happen, a point that echoes in the gambler’s fallacy, where past results are wrongly treated as relevant to independent trials.
- Aggregates are predictable, individuals are not. That is why a game operator can be confident about long-run averages while no player can be confident about the next result.
None of this offers a way to beat games designed with a built-in house edge. The mathematics was developed to describe uncertainty, not remove it. If gambling is causing harm for you or someone you know, a local support service can help.