Problem 18

Bertrand's Strange Three Boxes (1889)

Problem. There are three boxes, each with two drawers. Box A contains one gold coin in each drawer, box B contains one silver coin in each drawer, and box C contains one gold coin in one drawer and one silver coin in the other.

Solution. (a) All three boxes have the same chance of being chosen. So, the probability that box C is chosen is 1/3.

(b) Let C be the event “box C is chosen”, and similarly for events A and B. Let G be the event “a gold coin is found in one of the drawers from the chosen box”. Then

Using Bayes' Theorem,^{4} we have

# 18.1 Discussion

Although the name of Joseph Louis François Bertrand (1822–1900) is usually associated with the chord paradox (see Problem 19), his treatise Calculs des Probabilités^{1} (Bertrand, 1889) is a treasure-trove of several interesting probability problems. Of these, the box problem appears as the second problem in the book.^{2} However, the second part of the original question is in a slightly different form from that presented in (b) in the Problem. On p. 2 of his ...