[jonath_laurent]

Some excellent critiques have been provided in this thread already but I would like to offer a different perspective.

In programming terms, the switching argument (see original link) is incorrect because it does not typecheck. And it does not typecheck because variable A is used out of its scope when writing down the (5/4)*A expected value.

Indeed, variable A is tied to a specific random outcome and so it simply does not make sense to refer to it in an expected value ranging over this same outcome.

Trying to formalize the argument in a proof assistant makes the mistake clear and obvious. However, the ambiguity of the English language makes it possible for such typing errors to sneak in undetected.

[lisper]

> it does not typecheck ... because variable A is used out of its scope when writing down the (5/4)*A expected value

But that's simply not true.

> variable A is tied to a specific random outcome and so it simply does not make sense to refer to it in an expected value ranging over this same outcome

No, it isn't. It is the value of the chosen envelope -- which is known. You can open the envelope and look inside.

> Trying to formalize the argument in a proof assistant makes the mistake clear and obvious.

Have you actually tried this?

Consider the following alternative scenario: I give you $20 and offer you the following wager: we will flip a fair coin. If it lands heads you lose $10, tails you win (an additional) $20. Would you take that wager? If so, how is that different from the envelope problem? (Note that the $20 grant is a no-op, all it does is serve to fix the amount of the payouts after the coin flip.)