During the Renaissance, Galileo wrote about a question posed by a keen gambler, Cosimo II de' Medici, Grand Duke of Tuscany, whom he had tutored. Cosimo had noticed that when three dice were rolled, a total of 10 came up more often than 9. Even without watching real players as Cosimo did, we can observe the same phenomenon by simulating the game. We need only write a program and play the game virtually a thousand times (see the program in the box); the result will be close to the theoretical value, which can be proved.
The Tuscan paradox ----------------------
Yet there are just as many ways to express nine and ten as sums of three numbers between 1 and 6, which seemed contradictory to Cosimo! Galileo discovered the reason for this oddity, which has since been known as the Tuscan paradox. We can understand how it works by considering a game of heads or tails. If the coin is fair, the probabilities of heads and tails are both 1 / 2. If it is tossed twice, the four possibilities HH, HT, TH and TT are equally likely, so each has probability 1 / 4. If two coins are tossed at once, the probabilities of two heads and two tails are each 1 / 4, but the probability of one head and one tail is 1 / 2 because it combines the two cases HT and TH. Exactly the same principle applies to the Tuscan paradox: from this standpoint, the decompositions of nine and ten are not equivalent. The difference arises because nine can be decomposed into three equal numbers, whereas ten cannot.
Calculation shows that the probability of rolling nine is 25 / 216, whereas the probability of rolling ten is 27 / 216 (that is, 1 / 8). These figures show that Cosimo was a keen observer—and a truly dedicated gambler—since the probabilities differ by only 1 / 108.
When three dice are rolled, there are six distinct ways to obtain 9. Three involve three distinct values (1 + 2 + 6, 1 + 3 + 5 and 2 + 3 + 4), two involve two equal values (1 + 4 + 4, 2 + 2 + 5), and one has all three values equal (3 + 3 + 3). In the first case, six orders are possible, so each has probability 6 / 216 (note that 63 = 216, the total number of possible rolls). In the second case, only three orders are possible, giving 3 / 216. In the final case, only one is possible, giving a probability of 1 / 216. The total probability is therefore (18 + 6 + 1) / 216, or 25 / 216.