Two envelopes: one has twice the amount of the other. You pick one and see $X inside.
The other has $2X or $X/2, each with 50% probability
Expected value of switching: 0.5(2X) + 0.5(X/2) = 1.25X
So switching seems ALWAYS better... but that's impossible!
The fallacy: you can't simultaneously condition on your envelope being the smaller AND larger one while using X as a fixed value. The prior distribution matters.