A # Darling

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.