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.

For other paradoxes by Joseph Bertrand, see Bertrand's paradox (disambiguation).

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.

Choose a box at random. From this box, withdraw one coin at random. If that 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. 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.

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