You are given two envelopes. One has $X$, the other has $2X$. You pick one and see $Y$. The other envelope has either $Y/2$ or $2Y$. The expected value of switching is $(0.5 \times 2Y) + (0.5 \times Y/2) = 1.25Y$. Why is this logic flawed?

The Paradox: The mistake is assuming a uniform distribution over all possible values of X. In a real game, the expected values of switching and staying are always equal, as shown by the simulation.