Envelope Paradox: A simple solution?

[TD_WONDER]

1. I know there is X {the amount} in my envelope.
2. The probability that X is the smaller of the two amounts is 1/2, and also it is a 1/2 possibility that it's the larger amount also.
3. The other envelope may contain either 2X or X divided by 2 {1/2X}
4. If X is the smaller amount the other envelope contains 2X, BUT, If X is the larger amount the other envelope contains 1/2X
5. This means, the other envelope contains 2X with the probability 1/2 and 1/2X with probability 1/2
6. So the value of the money in the other envelope is:

{1\2} * {2X} + {1\2} * {1\2X} = {5\4}X

7. This is greater than X, so, on average, I gain money by switching
8. After the swap, I can denote that content of the other envelope {Y} and reason in exactly the same way as above
9. I conclude that the most rational thing to do is to swap back again
10. To be rational I will then end up swapping envelopes forever.
11. As it seems more rational to open just any envelope than to swap indefinitely we have a contradiction

[kentbball]

Originally Posted by kentbball

You should always switch.

Lets say there is a gameshow (bad gameshow but oh well) and the rules are as follows:

There are three amounts: 50$, 100$, and 200$.

Each round they give you the 100$ to start out with and they have another envolope that either contains 200$ or 50$.

So you start out with the 100$ envolope and your reasoning is that you will always gain more than you lose so you switch and get 200$ (WOOHOO). next round you switch and get 50$ (BOO). you than have 250$. So if you hadnt of switched at all you wouldve only had 200$. 200$ vs 250$ i know what i cwould like more.

probability says that it will always go 200-50-200-50-200-50 etc. It obviously wont but it will even out to that in the end. That means every two turns you will get an average of 250$ instead of 200$ if you decide not to switch.

No freakin clue how it works out to that but it just does.

nevermind im a dumbass. I just created a whole different question and made up an answer for it

[Patashu]

Why isn't anyone listening to me?

[Kit-]

Because your solution doesn't cause problems. We're trying to figure out why the other approach fails.
Plus you just stole it off Wikipedia.

[Patashu]

I didn't even look at wikipedia. And if my approach is true, how can the other one also be true? Problem solved?

[Laharl]

You also stand to lose more than you already have gained, which makes the arguement that you stand to gain more than you lose false.

I would keep the $100 because $200 is an obscure amount compared to an even $50, so I would figure that the gameshow would more likely put $50 in one and $100 in the other.

If it turns out I was wrong, then oh well. I still made $100.

If you always did things because you stood to gain more than you lose, then you'd always play at the dollar slots and spend more money than you'd make because hey, that dollar can turn into $5000 if you're lucky. It can also net you absolutely nothing.

Rule of gambling is that if you are ahead, stick with what you got because chancing a definate gain on a possible gain doesn't pan out more often than not.

[Afrobean]

The thing is, Laharl, that you have the same chance to gain or lose.

You have a 50% chance of getting 2x as much. You have a 50% chance of losing half of what you already had.

Since it's equal chances, it makes most sense to go with the better option, because it's not like some risky situation.

[The_Q]

Prisoner's Dilemma.

Nash Equilibrium.

Game Theory.

Econ.

Q

[Snuffles1012]

There is a solution to this...somehow...well when dealing with paradoxes you have to make up whatever you can think of that could give you a solution.

The game show picks two amounts of $, and one is twice the other (this is where we use are imagination) what if the money they picked was solid gold? then all you would have to do is look at or feel which envelope has more money thats how i slove this paradox.

[Patashu]

Originally Posted by Snuffles1012

There is a solution to this...somehow...well when dealing with paradoxes you have to make up whatever you can think of that could give you a solution.

The game show picks two amounts of $, and one is twice the other (this is where we use are imagination) what if the money they picked was solid gold? then all you would have to do is look at or feel which envelope has more money thats how i slove this paradox.

Except it's not gold.

I still seriously don't get what's wrong with my solution, though.

[Syntax Egon]

Originally Posted by Snuffles1012

what if the money they picked was solid gold? then all you would have to do is look at or feel which envelope has more money thats how i slove this paradox.

You, sir, win the Intelligent Poster of the Year of 2006 Award. To claim your reward, please pose with this piece of Uranium at groin level.

[Iwhaiwnfi]

..I really don't understand how this is confusing, or much less, a paradox.

If you had picked envelope B, you would have come to exactly the same conclusion. So if the above argument is valid, you should switch no matter which envelope you choose. But that can't be right.

Why not?

Say one envelope had 100, and the other had 50. Two situations:

1. You pick the one with 100 dollars. You immediately think that there are two possibilities. One envelope had 100, the other had 200, OR one had 100, and the other 50.

2. You pick the one with 50 dollars. You immediately think one had 100, and the other 50, OR one had 50, the other had 25.

The fact that you don't know the value of the envelopes allows different conclusions without it being a paradox.

[-Izzy-]

Originally Posted by RainGame53

Since you stand to win more than you stand to lose, you should switch.

Why?