Long before the famous correspondence between Pascal and Fermat of 1654, a physician and mathematician from Milan had already grasped the rules of chance at the gaming table.
Gerolamo Cardano (1501–1576) — the same man to whom we owe the formula for solving the cubic equation — was an inveterate gambler. Around 1564 he wrote the Liber de ludo aleae (“Book on Games of Dice”), which is regarded as the first systematic treatise on the calculus of probability. The work had an unlucky fate, however: it remained unpublished and appeared only posthumously in 1663, by which time Pascal and Fermat had already founded the new discipline, so that its historical influence was almost nil (Boyer; Katz).
In that little book Cardano stated, perhaps for the first time clearly, the idea that in a fair game the probability of an event is the ratio between the number of favourable cases and the total number of equally possible cases. Rolling a fair die, for instance, the probability of getting an even number is Cardano also understood that with a throw of two dice the possible cases are , and he correctly enumerated the outcomes — a piece of combinatorial reasoning by no means obvious at the time. He even glimpsed a rudimentary version of the product rule for repeated, independent events, though he made a few errors that only Pascal and Fermat would later correct rigorously.
Cardano’s figure reminds us that great mathematical ideas rarely arise in an instant: the “classical” probability attributed to Laplace has its roots in insights that matured over the two preceding centuries, often precisely in the concrete practice of gambling.
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Topics: Probability
Concepts: Probability · Favourable cases over possible cases
People: Gerolamo Cardano · Pierre-Simon de Laplace