Problem 15

Leibniz's Error (1768)

Problem. With two balanced dice, is a throw of 12 as likely as a throw of 11?

Solution. The sample space Ω = {(1, 1), (1, 2), . . ., (6,6)} is made up of 36 equally likely outcomes, where (a, b) are the numbers on the first and second dice, respectively, for a, b = 1, 2, . . ., 6. A throw of 12 can be obtained in only one way, namely (6, 6). However, a throw of 11 can be obtained in two ways, namely (5, 6) and (6, 5). Therefore, Pr{a throw of 12} = 1/36 but Pr{a throw of 11} = 1/18, and the latter is twice as likely as the former.

# 15.1 Discussion

The renowned German mathematician and philosopher Gottfried Wilhelm Leibniz (1646–1716) (Fig. 15.1) is usually remembered as the coinventor of the differential calculus with archrival Isaac Newton. However, he was also very much interested in probability.

Regarding the question in Problem 15 above, Leibniz states in the Opera Omnia (Leibniz, 1768, p. 217)

. . .for example, with two dice, it is equally likely to throw twelve points, than to throw eleven; because one or the other can be done in only one manner.

Thus, Leibniz believed the two throws to be equally likely, arguing that in each case the throw could be obtained in a single way. While it is true that a throw of 11 can be realized only with a five and a six, there are two ways in which it could ...