Actually, your betting strategy never used the $10 number in determining your formula. What you really did was use logic to show that a betting strategy that uses the first number (whatever it is, as you say) to buy the second envelope will work out in the long run. There's no $10 in that, except that your formula tells you that when you encounter the $10 subset, you should pay $10. But as soon as it does so, you no longer have any expectation of breaking even in that particular subset of trials! This last remark is crucial.

I see the issue. The key point is, you do not expect to have a breakeven strategy in any given subset, such as, 10 billion births!

Here's a better way to see it. Imagine that every intelligent being that has ever lived meets in the restaurant at the end of the universe. Someone says, "let's have all the people who were among the first 5% born to one corner of the room." Of course there's going to 1/20 of the attendees in the corner. Now they say "OK, can I have every being that was the 10 billionth born to raise their hand". Why would you expect 1/20 of those people to be in the corner? Generally, that would not be the case! The reason is, if you imagine that total populations are generated by some probability algorithm, then that algorithm must have some scale, a "median" birth number. The number 10 billion has some unknown relation to that scale, which generates the unknown correlations, just as the number $10 has some relationship to the envelope-stuffers scale! We used to think that because the relationship was unknown, Carter was an OK probabilistic argument. But what I'm now saying is, just because you don't know the correlation does not make the argument meaningful. You still don't pay $12.50 for the other envelope, and you still don't expect there to be 1/20 of the 10-billiionth-borns in the corner of the room! So Carter is just plain wrong, since it is an argument that completely involves that bunch of 10-billionth-borns. This is the subset I've been referrering to-- it's us!