Bertrand's Box Paradox
Introduction Bertrand’s box paradox, posed by Joseph Bertrand in 1889, is a paradox of elementary probability theory. According to the paradox, there are three boxes: A box containing two gold coins, A box containing two silver coins, and A box containing one gold and one silver coin. If a person were to choose a box at random, and from that withdraw a coin at random, which happens to be a gold coin, the paradox lies in the probability of the next coin drawn from the same box being a gold coin. At first glance, it may appear that the probability of the next coin being a gold coin is ½ as the next coin could either be gold, or silver. But it is in fact ⅔ as we will see below. Why is the Paradox Counterintuitive? After a box has been chosen, but before a coin is chosen, the probability of the box having two of the same kind of coin is ⅔. If the probability of ‘choosing a gold coin’, in combination with the ‘box having two of the same kind of coin’ is ½, then the probability of ‘choo...