Bertrand's box paradox

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:

1. a box containing two gold coins,
2. a box containing two silver coins,
3. a box containing one gold coin and one silver coin.

A coin withdrawn at random from one of the three boxes happens to be a gold. What is the probability the other coin from the same box will also be a gold coin?

A veridical paradox is a paradox whose correct solution seems to be counterintuitive. It may seem intuitive that the probability that the remaining coin is gold should be ⁠1/2⁠, but the probability is actually ⁠2/3⁠. Bertrand showed that if ⁠1/2⁠ were correct, it would result in a contradiction, so ⁠1/2⁠ 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.