4 ideas
People threw dice and astragali for thousands of years, yet no calculus of chance appeared until the 16th–17th centuries. The book argues that several barriers combined to delay it. Throws were used for divination, so the outcome was read as the will of a god rather than a regular frequency. Astragali were irregular bones that gave no obvious equally likely cases, and there was no combinatorial algebra or notation for counting outcomes.
Italian gamblers observed that a total of 10 came up more often than 9 with three dice, although each total can be made from six unordered partitions. Galileo resolved it by counting ordered outcomes instead of partitions. A total of 10 arises in 27 of the 216 equally likely permutations, while 9 arises in only 25. The case shows that the correct unit of equiprobability is the ordered outcome, and that gamblers' long experience detected a difference of about 1% before theory explained it.
In Liber de Ludo Aleae, Cardano called the full set of equally possible outcomes of a throw its 'circuit'. The chance of an event was the number of favorable outcomes compared against that set. He treated a wager as fair when the stakes were in proportion to the favorable and unfavorable cases. This framed probability as a ratio of counted equal cases, roughly a century before Fermat and Pascal.
In 1654 the Chevalier de Méré asked Pascal how to split the stakes fairly when a game of several rounds is interrupted before either player has won, known as the 'problem of points'. Pascal and Fermat worked it out in letters to each other. Fermat listed every possible continuation of the play, including rounds that would never actually be played. Pascal worked backward recursively, averaging what each player could expect at each position. Their answer was to divide the stakes according to each player's chance of winning, not according to the rounds already won, and this exchange is conventionally taken as the start of mathematical probability.