Bertrand's box paradox is a veridical paradox in elementary probability theory. It was first posed by Joseph Bertrand in his 1889 work Calcul des Probabilités. There are three boxes: a box containing two gold coins, a box containing two silver coins, a box containing one gold coin and one silver coin. The question is to calculate the probability, after choosing a box at random and withdrawing one coin at random, if that happens to be a gold coin, of the next coin drawn from the same box also being a gold coin. A veridical paradox is when the correct solution to a puzzle appears to be counterintuitive. It may seem intuitive that the probability that the remaining coin is gold should be , but the probability is actually . However, this is not the paradox Bertrand referred to. He showed that if were correct, it would lead to a contradiction, so cannot be correct. This simple but counterintuitive puzzle is used as a standard example in teaching probability theory. The solution illustrates some basic principles, including the Kolmogorov axioms. Solution The problem can be reframed by describing the boxes as each having one drawer on each of two sides. Each drawer contains a coin. One box has a gold coin on each side (GG), one a silver coin on each side (SS), and the other a gold coin on one side and a silver coin on the other (GS). A box is chosen at random, a random drawer is opened, and a gold coin is found inside it. What is the chance of the coin on the other side being gold? The following faulty reasoning appears to give a probability of : Originally, all three boxes were equally likely to be chosen. The chosen box cannot be box SS. So it must be box GG or GS. The two remaining possibilities are equally likely. So the probability that the box is GG, and the other coin is also gold, is . The flaw is in the last step. While those two cases were originally equally likely, the fact that you are certain to find a gold coin if you had chosen the GG box, but are only 50% sure of finding a gold coin if you had chosen the GS box, means they are no longer equally likely given that you have found a gold coin. Specifically: The probability that