The difficulty with these calculations is that they are for the “average” solution. Which, of course, only exists in the “long run.” We want the optimal solution for playing just once. See the full post.
Not exactly. The average is only the limit of the monetary gain function. The second method that I mentioned specifically deals with playing just once - it proves that for any M(A) and M(B) your likely gain is equal to your likely loss.
Specifically, you made your first pick and are holding an envelope in your hand. Your complete set of choices is (a) keep the envelope or (b) change it. To determine your most beneficial action you use the probability theory.
As I demonstrated earlier, each of those choices offers you a certain gain (positive or negative.) Such a gain would indeed converge to $150 (if $100 and $200 are in play) only if you do infinitely many tries. But we aren't calculating how much you'd win in a billion years! We only want to know what choice is most beneficial to you in each and every round of the game. Therefore it's not necessary to try one billion times, because advantages of one choice (if there are any) are present in every round of the game. For example, if you are playing Russian Roulette and given a choice of playing with a dummy round or a live round, the advantages of a dummy round are present each time you pull the trigger.
To illustrate this particular case: If you keep the envelope you have 50% chance that you get $100 and 50% that you get $200. If you change the envelope ... you have the same 50% chance that you get $100 and 50% that you get $200 (because the envelopes are identical and you haven't learned a thing from the fact that you held one in your hand.)
Since both options result in identical outcomes, the decision is irrelevant. Change or keep, you won't be any better off with any of those decisions. They are merely chairs on the deck of Titanic, they are of use only in a TV game, when the host wants to create an illusion of choice :-)
You can also think of it this way. Why do you say "you made your choice" when someone picks an envelope? Extend this phase; let the player to pick an envelope, then put it back (without looking), then pick another one ... for as long as you want. Once you are tired of picking you open what you have and get whatever money is inside. This way of seeing the problem eliminates the choice of "picking a different one"; it is truly a meaningless act in this scenario. Though if you wanted you could come up with a game that depends on such a choice - if, for example, each selection beyond the first one costs you some money.
Such riddles are easily solved by using standard mathematical methods, but without such a formal approach the solution may be not obvious. This is because humans depend a lot on their "gut feelings", small acts that are random and unjustified just because we aren't a donkey stuck between equally appealing food items.
This reminds me of a short story ("Chess at the bottom of the well" by Alexander Kazantsev) that I read recently. An Egyptologist discovers a well with a mathematical riddle. He has no clue how to measure the diameter of the well with just two dry reeds, but ancient candidates to priesthood had to do that to become priests - or to die trying.
His friend, a well educated guy, shows up, gets interested in the riddle and quickly solves it. He uses a quartic function to find the answer, and he is puzzled because these were not known in ancient Egypt. He pays no attention to this, though.
A parallel story line introduces a young candidate who is led into the well, the door sealed, and he is asked to find the solution or to die. He finds the solution by using a completely different approach.
This story is of interest here because modern math (of which what we used here belongs somewhere in sixteenth or seventeenth century) is a powerful set of tools that allow you to solve problems without trying to comprehend them. Most modern problems can't even be understood by a human mind, since they deal with infinities. Human mind is not a good tool to "guess" even a simplest surface integral, though a minimal toolset of calculus lets you do it just for fun :-)