# Bertrand's box paradox

Bertrand's box paradox is a problem, similar to the Monty Hall
problem, which was published by the French mathematician Joseph Bertrand
(1822–1900) in his 1889 text *Calcul des Probabilités*. Suppose
there are three desks, each with two drawers. One desk contains a gold medal
in each drawer, one contains a silver medal in each drawer, and one contains
one of each, but you don't know which desk is which. The question is this:
If you open a drawer and find a gold medal, what are the chances that the
other drawer in that desk also contains gold? This comes down, then, to
figuring out the probability that you've picked the gold-gold desk instead
of the gold-silver desk. Many people quickly jump to the conclusion that
there are two possibilities, and since the selection was random, it must
be 50-50. But this is wrong. Think of the initial selection as picking from
among six drawers:

before | ||
---|---|---|

S | S | G |

S | G | G |

1 | 2 | 3 |

after | ||
---|---|---|

G | G | |

G | ||

1 | 2 | 3 |

So, we have it narrowed down to three drawers, with an equal probability of each one being the one that was picked. One of the drawers is in desk 2, so there's a 1/3 chance that desk 2 was picked. Two of the drawers are in desk 3, so there are two 1/3 chances (i.e. a 2/3 chance) that desk 3 was picked.