identical envelopes meaning each one has the same # of bills?

 

Gambler's theory: You're betting $100 to win $200. Since your odds are approximately 1:1, the 2:1 payout makes it worth it. Switch.

Greed theory: More is better. Switch. If it fails, you should learn to not be a greedy bastard.

 

Guess i should have posted in teh other thread.

 

Still thinking about this one, but it seems to me the key is to focus on the scenarios where you keep your envelope. If you switch, there is an equal chance of gaining and losing. If you keep, however, you may have the high or low value but won't know.

I'm stuck with this being a value scenario. How much do you value what you have versus what you may get? The gambler in me shows it pretty clear that you should switch (bet $100 to win $300 half the time; showing about a 50% profit margin over the course of many bets). I guess I'll think about it on the way home.

 

Identical meaning that there is nothing in their outward appearance to suggest which contains more money. One could be red, I suppose, and the other green, as long as there's no way to tell which has more money before they're both opened.

The number of bills in each envelope may be the same or not.

 

Why "approximately"?

(And, while we're at it, it should be "because", not "since". )

 

"Since you have no way of knowing which envelope contains the larger sum, you pick one at random. The host asks you to open the envelope. You do so and take out a check for $40,000.

Here is where things get interesting, especially for contestants who know some mathematics.

The host now says you have a chance to change your mind and choose the other envelope. If you don't know anything about probability theory, particularly expectations, you probably say to yourself, the odds are fifty-fifty that you have chosen the larger sum, so you may as well stick with your first choice. (And you'd be right. But I'll come back to this in a moment.)

On the other hand, if you know a bit (though not too much) about probability theory, you may well try to compute the expected gain due to swapping. The chances are you would argue as follows. The other envelope contains either $20,000 or $80,000, each with probability .5. Hence the expected gain of swapping is

[0.5 x 20,000] + [0.5 x 80,000] - 40,000 = 10,000
That's an expected gain of $10,000. So you swap.
But wait a minute. There's nothing special about the actual monetary amounts here, provided one envelope contains twice as much as the other. Suppose you opened one envelope and found $M. Then you would calculate your expected gain from swapping to be

[0.5 x M/2] + [0.5 x 2M] - M = M/4
and since M/4 is greater than zero you would swap. Right?
Okay, let's take this line of reasoning a bit further. If it doesn't matter what M is, then you don't actually need to open the envelope at all. Whatever is in the envelope you would choose to swap. Still with me?

Well, if you don't open the envelope, then you might as well choose the other envelope in the first place. And having swapped envelopes, you can repeat the same calculation again and again, swapping envelopes back and forward ad-infinitum. There is no limit to the cumulative expected gain you can obtain. But this is absurd.

And there's the paradox. What is wrong with the computation of the expected gain from swapping?

The answer is everything. The above computation is meaningless - which is why it leads so easily to a nonsensical outcome. If you want to apply probability theory, you are free to do so, but you need to do it correctly. And that means working with actual probabilities, taking care to distinguish between prior and posterior probabilities. Let's take a closer look.

As with the Monty Hall Problem, if you really want to analyze the situation, you have to start by looking at the way the scenario was set up.

Let L denote the lower dollar value of the two checks. The other check thus has value 2L. Let P(L) be the prior probability distribution for the choice the host makes for the lower value in the envelopes. (This will affect the entire game. Of course, we don't know anything about this distribution. But we can see how it affects the outcome of the game. Read on.

When you make your choice

 

Are you discussing this problem, or the Monty Hall problem from the other thread?

In either case, it's not clear what you mean by "methods of choosing", and how you come up with four. Please describe them - on the appropriate thread.

 

seeing Old Man Monty removed the money from the initial envelope, it is therefore worthless. Trade it in!

 

SIIK2NR: Did you synthesize that analysis yourself, or did you find it somewhere? If the latter, where, may I ask?

(I ask because some of the syntax and spelling don't look like yours, not because I believe you incapable of this analysis.)