Bertrand's Box Paradox for Trading Interviews

Bertrand's box paradox shows that drawing a gold coin makes the other coin in the box gold with probability two thirds, not one half.

Bertrand's box paradox asks for the probability that the second coin in a box is gold, given that a randomly drawn coin from a randomly chosen box turned out to be gold, and the correct answer is two thirds, not one half. The result is a direct warning against the instinct to treat any two remaining possibilities as automatically equally likely.

There are three boxes. One box contains two gold coins, one box contains two silver coins, and one box contains one gold coin and one silver coin. A box is chosen uniformly at random, and then one of its two coins is drawn uniformly at random and shown to be gold. The question asks for the probability that the other coin remaining in that same box is also gold.

Label the boxes GG for the two-gold box, SS for the two-silver box, and GS for the mixed box. Before any coin is drawn, each box is chosen with probability one third. The probability of drawing a gold coin differs by box: from GG the probability of drawing gold is 1, from SS the probability of drawing gold is 0, and from GS the probability of drawing gold is one half, since exactly one of the two coins is gold.

The overall probability of observing a gold coin, averaging over the random choice of box, is one third times 1, plus one third times 0, plus one third times one half, which sums to one third plus one sixth, or one half. This is the denominator in the Bayes' rule calculation for the probability that the box is GG given that a gold coin was drawn.

The probability that the box is GG given a gold coin was drawn equals the probability of drawing gold from GG, times the prior probability of choosing GG, divided by the overall probability of drawing gold. That is one third times 1, divided by one half, which is one third divided by one half, or two thirds. Since the box being GG is exactly the event that the other coin is also gold, the probability the second coin is gold, given the first was gold, is two thirds.