The Relentless Two-Envelope Conundrum: A Paradox or Misapplication of Probability Theory

The primary objective of the paper is to make a case that the evaluation of the expected returns in the Two-Envelope Paradox (TEP) is problematic due to the ill-defined framing of X and Y as random variables representing two identical envelopes, where one contains twice as much money as the other. In the traditional literature, when X is selected, Y is defined in terms of the amount of money x in X using the values y= x and y=2x with equal probability .5, and vice versa when Y is selected. The problem is that the event X=x stands for two distinct but unknown values representing the money in the two envelopes, say $θ and $2θ. This renders X and Y ill-defined random variables whose spurious probabilities are used to evaluate the traditional expected returns. The TEP is resolved by applying formal probability-theoretic reasoning to frame the two random variables in terms of the two unknown values {θ, 2θ}, giving rise to sound probability distributions, whose expected returns leave a player indifferent between keeping and switching the chosen envelope.