It doesn't eliminate probability because it's the exact same envelope experiment. ;) In other words, there is still a 50% probability of gaining or losing. My p

It doesn't eliminate probability because it's the exact same envelope experiment. ;) In other words, there is still a 50% probability of gaining or losing. My post above isn't an argument, it's an explanation. The "undefined variable" is established by your own proposal, i.e., in #2 you first choose an envelope which has an amount we call "y". At that point "y" is undefined, because you specifically explain it could be either the higher or lower amount. I think your confusion lies in not considering carefully enough the meaning of your wording. In other words, you're performing a play on words to get a specific result, akin to the "heads I win, tails you lose" word play. Let me try to explain further below: Imagine actually doing this experiment with the two envelopes, opening the first and seeing $10. Wouldn't you say the other contains either $5 or $20? No. The other envelope contains the amount the person who prepared the envelopes placed in it. Try defining both values explicitly and this issue may become clearer. Imagine one envelope contains $10 and the other contains $5. Go through your above proposed equations using strictly those values. Remember, in a real experiment that would be completely valid.