Remarks on Pareto Distributions

Abstract

Let me remind you of the foundation of stochastic theory. The enumeration of the cases of equal probability, according to a saying of Poincaré, is no mathematical question, but a question of metaphysics. We remember those well-known three boxes, each composed of two drawers, each drawer containing one coin, gold or silver. The combinations are: gold in both drawers, box G G; gold in one and silver in the other, box G S; silver in both drawers, box S S. One box is chosen at random: it seems evident that the probability of having chosen the box GS is 1/3, one of three cases, equally possible a priori. When the box has been chosen, one drawer is opened and the coin inspected. It may prove to be a gold coin: in that case, there are two possibilities left. The other drawer may contain a gold coin or a silver coin. The probability of having chosen the box G S seems thus to be 1/2. The same consequence follows from the case of finding a silver coin in the opened drawer. We have found a paradox: by opening a drawer the probability was changed from 1/3 to 1/2.