Classic Problems of Probability

Problem 12

D'Alembert and the “Croix ou Pile” Article (1754)

Problem. In two tosses of a fair coin, what is the probability that heads will appear at least once?

Solution. Let H stand for heads and T for tails. Then the sample space when the coin is tossed is . Each of these sample points is equally likely, so applying the classical definition of probability

12.1 Discussion

Jean le Rond d'Alembert (1717–1783) was one of the foremost intellectuals of the 18th century and coedited the monumental Encyclopédie ou Dictionnaire Raisonné des Sciences, des Arts et des Métiers (Figs. 12.1 and 12.2).¹ However, d'Alembert was often at odds (no pun intended) with his contemporaries on questions of probability. On the above problem, he denied that 3/4 could be the correct answer. He reasoned as follows: once a head occurs, there is no need for a second throw; the possible outcomes are thus H, TH, TT, and the required probability is 2/3. Of course, d'Alembert's reasoning is wrong because he failed to realize that each of H, TH, TT is not equally likely. The erroneous answer was even included in his article Croix ou Pile² of the Encyclopédie (d'Alembert, 1754, Vol. IV, pp. 512–513). Let us examine the first part of his article (see Fig. 12.3):

Heads or tails (analysis of chances). This game which ...