Envelope Dilemma?

It doesn't make a difference because the IRS will get half of it anyway.

This is the classic "two envelope problem" in statistics.

There is no advantage whatsoever to swapping envelopes. It is challenging to prove mathematically (I think the formal proof is 14 pages long) but it is one of those "intuitively obvious" type problems. If envelope one contains n dollars and envelope two contains 2n dollars, the total "pie" is 3n dollars. If you select either envelope at random, your expectation is to make 1.5n dollars (.5 x n + .5 x 2n).

As Fez pointed out, swapping envelopes is no different than selecting the other envelope up-front. Absent any other information, both envelopes are worth exactly 1.5n. Opening an envelope gives you no additional information unless you have some intuition/hunch about the total size of the pie (i.e. you know a game show won't give our more than $1M).

You need to go research the problem and its solution. The paradox is that it is not beneficial to swap envelopes prior to opening but it is beneficial to swap after opening the first envelope.

If you want proof build a model in Excel where envelope 1 is worth a random number and envelope 2 has a 50% probability of containing 1/2 of the contents of envelope 1 and 50% probability of containing 2 times the contents of envelope 1. Run 100 trials and you will see that the cumulative values of envelope 2 will be ~125% of the cumulative values of envelope 1.

Baker, I don't know how to explain it any better than I already have. It is NOT TRUE that no matter how much money is in the envelope there is a 50% chance of the other envelope contains more. The problem doesn't say that. It says that one envelope contains twice as much as the other one. Your mistake is persisting in thinking that it is like a coin flip situation despite all the explainations why it IS NOT. Please look back at the explaination to sportsbettor's extreme example for the fallacy in your logic. Everybody else gets it. Then go say 100 Hail Mary's.

Rudy you just aren't very bright. For there not to be a 50% chance of other envelope containing twice the amount in the envelope you chose you would need assume that there is a greater than 50% chance that you originally chose the envelope with the greater amount of money. One of the iterations of this problem states that the person offering the envelopes is an eccentric billionaire. You are simply basing your assumptions on what you think is a lot of money. Mathematically it doesn't matter what you think is a lot of money as that doesn't effect the outcome.

I see it the way Baker does, but acknowledge I may be missing something. I'd like to hear the logic of Marilyn Vos Savant ( I think that's her name).

Oh, okay, massah. Well, smarter people than me have written academic papers on this very topic, and it might be useful for you to read one before shooting off your piehole. Here's a 9-pager from the astrophysics department at Cornell:

https://www.astro.cornell.edu/pdfs/orionnews/Two-envelope_Paradox.pdf

I've already mentioned most of the salient points that they talk about, but there's more detail and logic there.

Rudy, smarter people than you have asked me if I want fries with my value meal. Even the paper you cited concedes that the only time it may not be in ones best interest to swap envelopes is if the amounts you are dealing with are finite and known, at which point you can utilize probability theory to estimate the probability that the envelope you have chosen is the greater of the two.

From your posts I can surmise that you don't understand that there is a difference between what the paper you cited calls the Closed Envelope Problem (CEP) and the Open Envelope Problem (OEP). Do you think it is called a paradox because the solution is the same for both situations? No, it is called a paradox because while there is no benefit to changing envelopes prior to knowing the contents of the envelope you chose once you know the contents you should always swap (when the amount of money the offerer has is unknown, as it is in the problem david presented).

You also keep saying that the problem doesn't state that there is a 50% chance that the other envelope contains more. If it doesn't state that then what does is state? Please show me the math that supports your position.

I haven't read all the posts, so this may be redundant.

You have to know the moderator's "strategy" to analyze this question. Will he only offer you the
option if you pick the large envelope? Because he wants to "defeat" you? Or is the option
offered on each occasion? If it's Adam Corolla, I assume the worst.

This Envelope is the LAST Envelope on this Dilemma.