Order matters
Green 1 with copper 6 is a different throw from green 6 with copper 1. Treating the dice as different is what makes all 36 throws equally likely, so a total of 7 has six throws and a total of 2 only one.
45 / ANTIQUITY TO THE RENAISSANCE
Girolamo Cardano
With two dice, is an ace twice as likely as with one? Cardano counted the whole “circuit” of 36 throws and found 11 with an ace, not 12.
Explore the mathematics ↓
01 / THE CIRCUIT OF THROWS
Count every way the dice can fall
Throw two dice and find every throw with at least one ace.
Compare that with the “twice the chance” guess, then use Cardano’s rule to set a fair wager.
Bet on:
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THE IDEATwo fair dice have a circuit of 36 equally likely ordered throws. At least one ace comes up in 11 of them, not 12: the double ace belongs to both counts. Cardano’s rule lays the stakes in the proportion of favourable throws to the rest, 11 : 25. Divided by the whole circuit, his count is a modern probability: 11 ÷ 36.
The person behind the ideas
- Published Ars Magna, with solutions of cubic and quartic equations, in 1545.
- Wrote Liber de ludo aleae (Book on Games of Chance) in the 1560s; it was printed in 1663.
- Was a celebrated physician, and a lifelong gambler.
More of the mathematics
Algebra too
Ars Magna presented the solution of the cubic, crediting del Ferro and Tartaglia, and Lodovico Ferrari’s solution of the quartic.