> Since you seem to be making some authority arguments just know that I have a PhD in ML / stats and was top 30 in my country in olympiad level competitive maths so I DO know what I do and in this case with absolute certainty.

Let's review what you're claiming with absolute certainty. This is the first claim that I'm contesting:

> they say "one envelope contains 2A and the other A/2".

When pointed out that they actually don't say this, you clarified your point:

> the "writing" doesn't say it but the written formula at bullet point 7 says it. That's the mistake, the formula is wrong and does not describe reality.

The second part of that quote is correct: bullet point 7 has the mistake, the formula is wrong and does not describe reality.

The first part of that quote is incorrect: the written formula at bullet point 7 does not claim that one envelope contains 2A and the other envelope contains A/2. It claims that the first envelope contains A and the second envelope has a 50% probability of containing 2A and a 50% probability of containing A/2.

You correctly identify where the mistake is, but you misidentify what the mistake is.

If you read bullet point 6, it is very clear: "the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2". Notice that bullet point 6 does not claim "One envelope contains 2A and the other envelope contains A/2".

To illustrate my point, consider a simple coin toss game. If the coin comes up heads, you win $2. If it comes up tails, you win $0.50. How might we construct the EV calculation for this coin toss?

0.5 * $2 + 0.5 * $0.50

Notice how both parts of the formula refer to the same coin. They don't refer to different coins. You have one coin that you flip, and depending on how that one coin lands, there is some probability that you get $2, and some probability that you get $0.50. If you looked at that formula, you wouldn't think that the first part refers to different coin that the second part.

Similarly, here we have one envelope (similar to having one coin) and we have uncertainty about the what the envelope contains (similar to having uncertainty about how the coin will land). We have 50% chance that the envelope will contain 2A, and we have 50% chance that the envelope will contain A/2. Thus, we arrive at the (incorrect) formula:

0.5 * 2A + 0.5 * A/2

It's very obvious that both parts of the formula refer to the same envelope, just like in the coin toss formula both parts of the formula refer to the same coin. To be extremely clear, I am not claiming that the formula is correct. I am claiming that both parts of the formula refer to the same envelope, not to different envelopes. To be specific, I am refuting this claim that you made:

> they say "one envelope contains 2*A and the other A/2".

They don't say that.

Furthermore, you make an additional incorrect claim:

> The right formula is changing either the 0.5A by A or the 2A by A.

Here you are just randomly changing the formula in order to make it output 0EV (which is the correct answer). Just because you get a correct answer does not mean your computation was correct. In this case it is incorrect, because A was defined badly. In order to fix the formula you would have to fix the definition of A. If you keep the incorrect definition of A and just shuffle symbols around until you get the correct result, your computation is still incorrect.