Envelope Dilemma?

This is a link to another site the Wizard of Odds has. It's problem number 6:

https://www.mathproblems.info/group1.html

With all due respect David....The wizards answer is simply mental masturbation. I have read it upteen times by now.

Overthinking, Underthinking. We get 2 very viable answers. And the wizards rate of increase vs. amount of increase argument notwithstanding.

The variables are indeed simple and it doesn't matter what amount is even in the 1st envelope, we can call it X. If you switch you will get 2X or X/2. If you stay you will have X. Only if you knew already that it was worth X. Otherwise without even looking in the envelope the choice to switch would be moot.

Needless to say this has probably provided great thesis material for many a graduate students.

The conclusion of paradox implies not right answer. Or to take a message from DanielSong39, a poorly defined mathematical problem with no solution.

This is very simple.

You are betting $25,000 to win $50,000 on a coin flip.

I'd take that all day long.

Take it to an extreme to better understand.

One envelope has 100 times more money than the other.

The envelope you have contains $1000.

Do you take an even chance on switching when you will get either $10 or $100,000?

It's not a friggin' coin flip! Sheesh. Read the preceeding posts. It is a predetermined outcome.

Let's take your extreme example, sportsbettor: what if instead of $1000, the envelope you opened had the $100,000? Are you going to swap that one too in hopes of getting $10 million? At what point do you step back and say, "it isn't logical that the other envelope might contain 100 times more than this one?"

If anyone still believes that it is correct answer is always to swap, then why even bother looking in the first envelope you chose?

That is a complete solution. (Final Answer: Doesn't matter if you swap or not.)

JoeOffsuit

Rudy it is a coinflip situation. After you choose an envelope there is a 50% chance that the other envelope contains half as much and 50% that the other envelope contains twice as much. You are making assumptions and trying to make "logical" choices when you don't have the proper information to base these assumptions.

Good point Joe, just pick the other envelope to begin with.

There is no logical basis for assuming that there is a 50% chance that the other envelope contains twice as much money...

This is a classic "shot in the dark" problem. Who knows what's inside that black box?

Daniel-

The problem states that one envelope contains twice as much money as the other envelope. So there are two chocies, one contains X and one contains 2X. If there is a 50% probability that the first envelope chosen contains X then there has to be a 50% probability that the second envelope contains 2X.