Problem 1

Cardano and Games of Chance (1564)

Problem. How many throws of a fair die do we need in order to have an even chance of at least one six?

Solution. Let A be the event “a six shows in one throw of a die” and its probability. Then . The probability that a six does not show in one throw is . Let the number of throws be n. Therefore, assuming independence between the throws,

We now solve obtaining , so the number of throws is 4.

# 1.1 Discussion

In the history of probability, the physician and mathematician Gerolamo Cardano (1501–1575) (Fig. 1.1) was among the first to attempt a systematic study of the calculus of probabilities. Like those of his contemporaries, Cardano's studies were primarily driven by games of chance. Concerning his gambling for 25 years, he famously said in his autobiography (Cardano, 1935, p. 146)

. . .and I do not mean to say only from time to time during those ...