On the Two-envelope Paradox

Abstract

The two-envelope paradox is explained from the standpoint of the axiomatic construction of probability spaces. The probabilities of events and the numerical characteristics of random variables should be calculated after constructing the probability space. If we correctly define the space of elementary outcomes Ω, correctly set the probabilities of elementary outcomes if Ω is a finite or countable set, correctly determine the distribution laws of the random variables under consideration, if Ω is a set of continuum, the paradox disappears. For each probabilistic space, a different answer is obtained to the question of which of the players benefits from the exchange of envelopes, or whether this exchange does not bring benefits to any of the players.