re: The Two Envelopes Problem (A Monty Hall Problem Variant)

The envelope is the boy's mother.

Right but the problem states it is bullshite / paradoxical - OP just failed to post the actual challenge being proposed.

I would trade both envelopes for the box that has the cat that is either alive or dead

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Which is why the whole exercise is BS, even the 3 option question.

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The original Monty hall problem is different in that you gain additional information after your initial choice, which is why it is advantageous to swap.

Not the case here.

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I'm just using logic, common sense, and rational thinking which tells me this is bullshite

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That’s kind of the point. The orginal Monty Hall problem defies common sense.

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So I'm supposed to go out of town on Friday morning to Raleigh and play in a golf tournament for one day. One of my daughters has been sick (24 hours of... Digestive distress) but is better. My wife is suffering from the same symptoms right now. And gasoline is difficult to find.

What should I do? Am I going to get sick? Will I find enough gas to get up there playing the tournament and come back on Saturday? Additionally we are leaving Sunday afternoon for about a 10-day vacation as a family driving north.

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You're probably going to do poorly in the golf tournament anyway. Leave and park the car at a Wal-Mart in the next town over. Get a trusted buddy to drive you back home and sneak down into the basement. Stay down there watching movies and eating shitty junk food you bought at the Wal-Mart, keeping yourself isolated while your family doesn't even think about coming down and busting you because they're upstairs counting the tiles in the bathroom as they're whiling away the hours getting the bug out of their systems.

Sneak back out the next morning and have the same buddy pick you up and drive you back to your car, at which point you'll follow him to his house where you'll shower and shave to be presentable when you return home after your "golf trip", having saved yourself travel time, money, and the frustration of golf. Then you pick up your no longer leaking (or contagious) family and go on vacation.

I don't know why this is so hard to figure out.

Nalebuff asymmetric variant
The mechanism by which the amounts of the two envelopes are determined is crucial for the decision of the player to switch or not his/her envelope.[9][10] Suppose that the amounts in the two envelopes A and B were not determined by first fixing contents of two envelopes E1 and E2, and then naming them A and B at random (for instance, by the toss of a fair coin[11]). Instead, we start right at the beginning by putting some amount in Envelope A, and then fill B in a way which depends both on chance (the toss of a coin) and on what we put in A. Suppose that first of all the amount a in Envelope A is fixed in some way or other, and then the amount in Envelope B is fixed, dependent on what is already in A, according to the outcome of a fair coin. ?f the coin fell Heads then 2a is put in Envelope B, if the coin fell Tails then a/2 is put in Envelope B. If the player was aware of this mechanism, and knows that they hold Envelope A, but don't know the outcome of the coin toss, and doesn't know a, then the switching argument is correct and he/she is recommended to switch envelopes. This version of the problem was introduced by Nalebuff (1988) and is often called the Ali-Baba problem. Notice that there is no need to look in Envelope A in order to decide whether or not to switch.

This seems to be more a more interesting variant - guess this one is actually self evident though. If given a known amount, would you flip a coin risking half the money with chance to double up.

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Problem
Basic setup: You are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it contains. You pick one envelope at random but before you open it you are given the chance to take the other envelope instead.

The switching argument: Now suppose you reason as follows:

I denote by A the amount in my selected envelope.

The probability that A is the smaller amount is 1/2, and that it is the larger amount is also 1/2.

The other envelope may contain either 2A or A/2.

If A is the smaller amount, then the other envelope contains 2A.

If A is the larger amount, then the other envelope contains A/2.

Thus the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2.

So the expected value of the money in the other envelope is:

(1/2)(2A)+(1/2)(A/2) = (5/4)A

This is greater than A so, on average, I gain by swapping.
After the switch, I can denote that content by B and reason in exactly the same manner as above.

I will conclude that the most rational thing to do is to swap back again.

To be rational, I will thus end up swapping envelopes indefinitely.

As it seems more rational to open just any envelope than to swap indefinitely, we have a contradiction.

The puzzle: The puzzle is to find the flaw in the very compelling line of reasoning above. This includes determining exactly why and under what conditions that step is not correct, in order to be sure not to make this mistake in a more complicated situation where the misstep may not be so obvious. In short, the problem is to solve the paradox. Thus, in particular, the puzzle is not solved by the very simple task of finding another way to calculate the probabilities that does not lead to a contradiction.

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I would trade both envelopes for the box that has the cat that is either alive or dead

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NEVER pay money for some dead cat in a box. It's cheaper and more rewarding to make your own dead cats at home.

From my understanding the paradox works when there are two people and two envelopes (or objects of value). In this scenario each person has an advantage to switch, but the overall act is illogical and thus paradoxical.