# Mises, Probability, and the Two Envelopes Problem

In Human Action, Mises distinguishes between what he calls “class probability” and “case probability.” He defines class probability as such:

Class probability means: We know or assume to know, with regard to the problem concerned, everything about the behavior of a whole class of events or phenomena; but about the actual singular events or phenomena we know nothing but that they are elements of this class.

This is the ordinary sort of probability. We reach into an urn containing seven red balls and two white balls, so the probability of choosing a red ball is 7:2. We can say this because we have knowledge about the class of balls in the urn. Mises distinguishes this from case probability:

Case probability means: We know, with regard to a particular event, some of the factors which determine its outcome; but there are other determining factors about which we know nothing.

Mises goes on to criticize the tendency to conflate case probability with class probability. To say that a political candidate’s odds of winning an election are 9:1 is a meaningless statement; there is no class of ten elections of which nine result in the candidate’s victory. At best, it is a faulty analogy.

There is a famous math problem that demonstrates the error in applying the reasoning of class probability to case probability: the Ali Baba problem (also known as the two envelopes problem)