I think you are entirely right, that people don't visualize.

Lady Average may not be as good-looking as Lady Luck, but she sure as hell comes around more often.

Anonymous

Isn't there an additional requirement that there is a minimum element in the set?

I believe you're thinking of this blag post.

There are two envelopes in which I, the host of the game, put two different natural numbers, chosen by any distribution I like, that you don't have access.

Now, before everything I said happens, you must devise a strategy that guarantees that you have a greater than 1/2 chance of winning.

Well natural numbers and simple greater than satisfying makes it easy. "If one THEN swap ELSE keep."

Actually there are no uniform distribution in this set (an infinite enumerable set).

That is what I was getting at with 'ruled out of being distributions'.

I don't see your conclusion holding. I am inclined to say: Therefore there are no distributions which that, regardless of how much money you find in envelope A, envelope B still has an equal chance of being twice as much or half as much.

I suppose "numbers selected from all the numbers in the series 2^n" and so forth are ruled out of being distributions based on the "infinities and uncomputable things are just silly" principle? (I am fairly confident that) something on that order of difficulty is going to required to provide the envelopes. A task that is beyond even Omega in the universe as we know it but perhaps not beyond an intelligent agent in the possible universes that represent computational abstractions natively.

There is another very cool puzzle that can be considered a followup which is:

There are two envelopes in which I, the host of the game, put two different natural numbers, chosen by any distribution I like, that you don't have access. The two envelopes are indistinguishable. You pick one of them (and since they are indistinguishable, this can be considered a fair coin flip). After that you open the envelope and see the number. You have a chance to switch your number for the hidden number. Then, this number is revealed and if you choose the greater you win, let's say a dollar, otherwise you pay a dollar.

Now, before everything I said happens, you must devise a strategy that guarantees that you have a greater than 1/2 chance of winning.

Some notes:

1- the problem may be extended for rational, or any set of constructive numbers. But if you want to think only in probabilities this is irrelevant, just an over formalism.

2- This may seem uncorrelated to the two envelopes puzzle at first, but it isn't.

3- I saw this problem first on EDITthis post on xkcd blag. Thanks for Vaniver for pointing out.

Another commentary I once read regarding the "two envelopes paradox":

In order to make any sense out of this problem, you have to assume some prior probability distribution over the amount of money in the two envelopes. For many of these possible distributions, once you open one of the envelopes and learn how much money is inside, you now know more about whether the other envelope has more or less money than the one you opened. For example, if you assume that neither envelope has more than $1,000 and then open an envelope with $800, the other envelope has to have $400, so, contrary to the line of reasoning in the "paradox", switching would be bad.

On the other hand, perhaps you only want to think about distributions for which it seems the paradox still holds: ones in which that, regardless of how much money you find in envelope A, envelope B still has an equal chance of being twice as much or half as much. Well, it turns out that you can prove that this criterion also implies that the expected value of the amount of money in envelope A is infinity. This makes the paradox seem much less paradoxical: first, when your expected value is infinity, any specific finite result is disappointing (which is why switching is always correct), and second, any finite number multiplied by infinity is still infinity (which explains how each envelope can have an expected value of 1.25 times the other).

(Apparently, my original source was David Chalmers, of all people.)

I used to be a frequentist, and say that the probability of the unfair coin landing heads is either 4/5 or 1/5, but I don't know exactly which. But that is not to say that I saw probabilities on things instead of on information. I'll explain.

If someone asked me if it will it rains tomorrow, I would ask which information am I supposed to use? If it rained in the past few days? Or would I consider tomorrow as a random day and pick the frequency of rainy days in the year? Or maybe I should consider the season we are in. Or am I supposed to use all available information I have? The latter I would call subjective probability. If someone instead passed me the children problem I would say 1/3 because this problem implicitly tells me to consider only the what tells the enunciate.

But simply asking for the probability without a context, I would say either that this is a no question, i.e. that the enunciate is imprecise and lacking information, or I would believe that the interrogator was asking for a intrinsic probability, in which case I would say either 0 or 1, but I don't know which.

But I did believe in intrinsic probability, in some cases, like quantum mechanics.

This view of mine became hollow after I started inquiring myself about this intrinsic probability. Even if such a thing existed, it couldn't be differentiated from what I called subjective probability. By Occam's razor I shouldn't create 2 kinds of probabilities that I cannot tell apart. This thought was partly inspired by reading lesswrong, not a particular post, but by seeing the ease in which what I called subjective probability was used in several occasions.