Quote:AsswhoopermcdaddySuppose I were Satan and offered you those three envelopes, but instead called the prizes: Hell, Purgatory, and Heaven...

I think I would prefer purgatory to heaven anyway, so I'd stick with that. Along the lines of gambling on eternity, I predict it is just a matter of time before somebody brings up Pascal's Wager -- hopefully in a separate thread.

Quote:weaselmanA set of all possible finite combinations of words is isomoprphic to the set of all integers (or rationals).

Yes, every rational can be decomposed as a product of Pi^ei for some p's and e's (e's can be negative for rationals). Treat the primes as the atomic words. So what? That's just the fundamental theorem of arithmetic. But, even then you can't describe them. The p's and e's themselves get too large to describe. And using your logic, the e's would also need to be similarly decomposed. Because of the problem describing the e's, the description you mention is "circular" in its logic.

But hey, you're pretty smart 8-)

--Dorothy

Quote:DorothyGaleThe p's and e's themselves get too large to describe.

No, you can describe any integer in a finite number or words.

Quote:But hey, you're pretty smart 8-)

Yeah, I know, but thanks anyway :)

Quote:weaselmanNo, you can describe any integer in a finite number or words.

Now, does this help in any way in picking an integer at random?

--Dorothy

Quote:DorothyGaleNow, does this help in any way in picking an integer at random?

--Dorothy

Nope. You said yourself, it was off topic :)

Quote:WizardI think I would prefer purgatory to heaven anyway...

Excellent choice.

First, I would say that the obvious solution is to grab both envelopes and run like hell.

Assuming that's not a possibility, I'm afraid that I don't understand why there is a problem in the expected value of switching being 125%. I'm going to toss out the concept of infinitely switching envelopes, since the problem does not seem to indicate that this is a possibility. You get one and only one switch. Instead, I guess I look at it as either a game show concept, or maybe from the point of view of a casino. Let's run the game 100 times with 100 different people, and give them all the same choice, and assume a 50% probability of the envelope they switch to being twice as much and a 50% probability that it is half as much. Again, assume that their initial pick is an envelope of $100.

If all the people switch, then 50% of them will end up with $200, and 50% of them will end up with $50. The casino running the game will have to pay out $12,500 in total, or an average of $125 per person. I might assume that half of the people, not understanding the mathematics behind the game, would choose to keep the initial envelope. In other words, 50 people end up with $100, 25 people end up with $200, and 25 people end up with $50. All told, the casino would have to pay out $11,250, or an average of $112.50 per person.

I think, mathematically, everyone agrees that the expected value is 125% and thus, mathematically, the switch should be made. The only other problem I can think of that people have is that it doesn't seem logical to switch the envelopes. However, I contend that it is logical. Mainly because the concept of doubling your money greatly outweighs the concept of halving your money. The reason for this, at least in my mind, is that doubling your money is a lot harder to achieve than halving your money.

Let's say you took $100 and put it into some sort of compound interest bearing account earning 5% per unit of time. It would take 15 such units of time to reach $200. On the other hand, if it lost 5% per unit of time, compounded, you would be down to under $50 in 14 such units of time. Take out compounding, and you reach $200 in 20 units ($5 gained per unit), or you are down to $50 in 10 unit ($5 lost per unit).

Think of gambling, where you start with $100 and your stop goals are either $50 or $200, whichever comes first. You're going to play $5 a hand blackjack. Obviously, it's a lot easier to lose $50 than it is to win $100. But let's say you take a completely fair game, like flipping a fair, non-biased coin. Heads you win, Tails you lose. Wouldn't math show that the odds of getting 10 Tails in a row far outweigh getting $20 Heads in a row? (And yes, I know that there may be other up and down movements in there, but the point remains that it's a lot easier to go down by 1/2 then it is to go up to twice as much.)

Thus, from MY logical point of view, the chance that the envelope you switch to could contain twice as much money as the envelope you just opened, seems like a great choice, given how difficult it would be to take the money in your hand and double it.

P.S. And through the first few pages of reading, I was thinking of Pascal's Wager, but since the Wizard has forbidden it in terms of this problem, I left it out of my ramblings. :)

Quote:WizardYour rant on politics is so significantly off topic that it should have been made into a separate thread. Please copy and paste it into a separate thread if you wish this to remain on the board.

Relax, Wizard. Blow it away yourself if you want. I thought it was appropriate because the "problem" that is the subject of this thread isn't a problem at all, and the fact that it seems to be a conundrum is merely an artifact of our general inability to understand some of the basic dynamics of monetary interchange. This, in turn, leads us to believe in nonsensical promises re the economy made by our beloved leaders.

And allow me to clarify two definitions:

"Statement": A verbal or written expression that a given person agrees with or feels neutral about

"Rant": A verbal or written expression that a given person disagrees with

Quote:DorothyGaleNow, does this help in any way in picking an integer at random?

--Dorothy

For the fun of it, here is an algorithm to pick any integer at random.

1. Toss a coin. If heads, write down 0, else write down 1.

2. Toss a coin again. If heads, go to #1 else go to #3

3. Take the sequence of zeroes and ones you generated, treat is as an integer in binary notation and convert to decimal.

4. Toss a coin again. If heads, stick a minus sign in front of the number

This will generate a fairly steep distribution with a peak around zero, but you can play with #2 to make it as wide as you like (e.g, keep tossing the coin until you see ten heads in a row to stop).

Quote:konceptum...I think, mathematically, everyone agrees that the expected value is 125% and thus, mathematically, the switch should be made...

In this forum, so far, you and TheNightfly are the only ones to firmly take a stand that the 125% argument is in fact correct. I think everyone else can see that if you know you are going to switch to the other envelope, then you may as well pick the "other one" to begin with. The question is where is the fallacy in logic of the 125% argument. See also my response to the TheNightfly post.

Also, I don't forbid talking about Pascal's Wager, just make a new thread for it.