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

The Two-Envelope Problem (TEP) is revisited to argue that the standard evaluation of expected returns relies on spurious probabilities arising from a misuse of formal probability theory. The source of the problem is the ex post framing of two identical envelopes, X and Y, one containing twice as much money as the other, after one envelope, say X, has been selected and its content X=x observed. The value x is then used to define Y in terms of the values y=x/2 and y=2x, each assigned probability .5, with an analogous derivation when Y is selected. This renders X and Y ill-defined random variables because the relevant probabilistic framing must instead be based on the original experimental setup, prior to any selection or observation, where the envelope contents are unknown, say $θ and $2θ. Framing the original setup using axiomatic probability, the dependence between X and Y is accounted for when x=θ, y=2θ, and when x=2θ, y=θ. The ensuing joint distribution of X and Y determines that the expected returns imply indifference between keeping the chosen envelope and switching, explaining away the ‘paradox’ as a misapplication of probability theory.