Galileo’s passe-dix game. Passe-dix is a game of chance played with three fair dice. Players bet whether the sum of the faces shown on the dice will be above or below 10.

During the late 16th century, the astronomer and mathematician Galileo Galilei was asked by the Grand Duke of Tuscany to explain why “the chance of throwing a total of 9 with three fair dice was less than that of throwing a total of 10.” (Interstat, Jan. 2004). The Grand Duke believed that the chance should be the same, since “there are an equal number of partitions of the numbers 9 and 10.”

Find the flaw in the Grand Duke’s reasoning and answer the question posed to Galileo.