Bertrand's box paradox

For other paradoxes by Joseph Bertrand, see Bertrand's paradox (disambiguation).
The paradox starts with three boxes, the contents of which are initially unknown

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A coin withdrawn at random from the three boxes happens to be a gold coin, then what is the probability that the next coin drawn from the same box is also a gold coin?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.[1] 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.

  1. ^ "Bertrand's box paradox". Oxford Reference.