I’ve been coming at this from several points of view, so I’ll post what I have and will await your response. I hope this all makes sense.

I want to convince you that only one trial of anything is useless in determining odds/outcomes. You must always have multiple trials. In every case that odds are determined, multiple trials are involved. Those trials must be defined precisely. In the course of those multiple trials, something must repeat. That something also must be defined precisely. If something doesn’t repeat, then no odds can be calculated on it. The problem with the 2-envelope game is that it always contains an amount that doesn’t repeat. It seems like this isn’t true, but it is. That’s why it’s confusing.

You wrote, “every trial offers a chance to go up or down a fixed amount (fixed for the trial).” It doesn’t matter that an amount can go up or down if it doesn’t ever go up or down in that game no matter how many times it is played. In all of the 2-envelope games every played or conceived, some amounts have never gone up and down! In fact, if you consider that every amount put into an envelope is a part of the game because it can be drawn, then you can play a billion trials, a billion games, or a billion combinations of those, and some amount still won’t have gone up and down. You can play an infinite number of trials, games, or combinations thereof and there will still be an amount that didn’t go up and down. What happens to the TEP logic for those amounts? What difference does it make that an amount could go up or down when it never does?

Here’s another way to respond to your statement above: so you must average the number of times it goes up with the number of times it goes down. How do you do that when, in theory, every amount doesn’t go both up and down, even though, in fact, both of those amounts exist?

You might say that every amount drawn has gone both up and down, but to consider your odds, but that’s simply luck that that number was drawn twice before twice its amount was drawn once. That luck can’t be used to calculate odds. You must consider the odds for every amount put into an envelope during any game because there is some non-zero change that every envelope will be chosen.

Below, I’ll explain in a different way, but I think that having you actually play a game and show me how the logic applies to it will be the most instructive.

Odds are always based on multiple trials with something repeating and something (the outcome) varying. It is always evident what constitutes a trial and what repeats. In fact, these must be precisely determined in order to calculate odds.

Here’s an example: I want to calculate the odds of a coin coming up heads when I toss it. I don’t know if the coin is fair. So, I toss the coin repeatedly (that’s what repeats) and average the outcomes (that’s what varies). The average of the outcomes will get closer to the answer with repeated trials.

Suppose I shuffle a deck of cards and then remove the top 10 cards. I want to figure out the odds that two pair will win against every other possible hand. I can do this by playing many hands (that’s what repeats) and recording the number of times 2 pairs wins (that’s what varies). Once again, the average of the outcomes will get closer to the answer with repeated trials.

This is always how odds are calculated. Sometimes, math can aid, but sometimes it can’t. For example, in the coin problem above, no math will suggest an answer. Only doing the experiment or analyzing the physical makeup of the coin will suggest an answer.

If an experiment can’t be described in a similar way, then odds cannot be calculated.

So, I’d like you to play the 2-envelope game several times for me (i.e. write out what you did and tell me the outcome) and then show me how you applied the TEP logic to the game and then calculated the odds. You can just play 3 or 4 trials and then show my how the odds apply.