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Agreed, this reminded me of the math book I read as a kid (sorry, don't remember the name anymore), which asked a seemingly simple question: "is a random triangle acute"

The book then went over a few different methods of constructing random triangles (pick 3 random points, pick 2 random angles, pick 3 points on circle, etc..) and shown how each approach would give a different answer to the problem.

This reminds me of a discussion I once contributed to in an AI research lab, which I got to participate through a Research Experiences for Undergrads program. The topic was genetic programming and someone casually mentioned picking a genome (code) at random. They seemed to assume that would create an initially flat distribution. Whoa hold up there, I said, you can't assume that. They asked, what did I mean?

I told them that both the way the "random" genome bits are picked and the way the genome bits are interpreted (fitness function) changes the distribution. For example, if the genome bits are picked as a random integer between 0 and 2^n-1, but the fitness function is a popcount of 1 bits, then the distribution will be hill shaped for the same reason that a 7 is more likely to be rolled with two dice than a 2 or 12.

Out of the researchers and undergrads there, I think about half nodded their heads as soon as I pointed it out, and the other half struggled to understand or accept the idea that interpretation could affect probability distribution, no matter how much exposition or supporting arguments I gave.

I didn't know about the water/wine paradox then, I wish I had. But now I wonder if part of the water/wine paradox (the fact it is even considered a paradox) has to do with that bi-modal distribution I saw between people who understood that interpretation affects probability distribution and those who could or would not accept that.

The problem is that "uniform distribution" is hard to define. An even distribution of water/wine values is not the same as an even distribution of wine/water values.

Thinking in terms of possible states, you can choose an even distribution of x from 0 to 1 and define:

water/wine = x/(1-x)

wine/water = (1-x)/x

(which avoids being water-biased or wine-biased)

So this is defining the random variable not as the ratio but as the implicit function that computes the ratio?

So we are really looking at the density function of the hidden variable X?

For the original problem it feels like the interval constraints on the ratio are causing the issue since they will be inverse. But maybe I am not clear on that...

The problem is not completely specified. However, in the study of probability and statistics there is a tendency to charge ahead and offer answers to incompletely specified problems, using e.g. the Principle of Indifference. The wine/water paradox illustrates a flaw in that approach. The Two Envelopes problem illustrates another flaw.

The "Grandfather" in the story argued that we can consider the problem specified if we assume that "the set of all distinct possible states" are uniformly distributed -- which is reasonable and correct when solving problems in quantum mechanics but unreasonable in ordinary everyday situations.

Thinking along those lines, you can construct a set of all possible concentrations: (0 wine particles, N water particles), (1 wine particle, N-1 water), (2, N-2), ... (N, 0). That led to my x/(1-x) formulation, which has the advantage of avoiding the paradox as stated, but I would not argue it's a well-founded result.

Why is it not well founded? Here an abstract amount (so a state) seems more realistic measure than the ratio which is a function of wine and water.

For one thing, there could be many distinct quantum states which correspond to the same water/wine ratio, so uniformly-distributed quantum states might not correspond to uniformly-distributed concentrations.

But more fundamentally, it seems a stretch to apply these quantum mechanical assumptions. One could interpret the question as "in all possible real-world scenarios in which such a test might be performed, how often would the concentration be in this range?" ... in which case one could consider all realistic models for how the mixture might have been created, and the likelihood of each of those actually being the case, and consider more and more things coming up with a more complete answer ... but one would never have a correct answer without actually stating some additional assumptions.

> water/wine = x/(1-x)

> wine/water = (1-x)/x

It's symmetrical but not "naturally" uniform. If we consider both quantities as independent and uniformly distributed over [0..1[, then 1:1 ratio becomes much more likely to occur than 1:1000000. I find that approach "the most natural". The probability P(water/wine < 2) is 7/8 in that case.

I agree. There are a number of ways to model the process of creating the mixture that will avoid the water vs. wine bias that is the essence of the "paradox" as stated.

The main lesson is that the Principal of Indifference is "not even wrong". (It's nonsense.)

This seems just like Bertrand's Paradox

https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)

That is, given a random chord of circle, what is the probability that it's longer than X? [1]

It's the same issue: how are you randomly choosing the chord? Random (uniformly chosen) intersection point with the radius? Two random (uniform) points on the circumference? People likewise make assumptions that, they argue, pin down how you have to choose the chord and derive solutions on that basis.

[1] In the official version, X = "side length of an inscribed equilateral triangle", but any value could raise the same issues.

The question is more about what kinds of distributions are in some sense "reasonable" for this setup. A uniform distribution for wine/water is problematic because it implies a non-uniform distribution for water/wine, and it is reasonable to expect that these distributions should be the same.

The underlying problem is still underspecified, and there is a class of distributions which have the appropriate symmetry, as described in the paper. The range of possible answers is still more limited under this symmetry assumption (that the ratio of water/wine and wine/water should have same distribution) than under no assumption (in which case any probability between 0 and 1 is acceptable).

For me that paradox just means that you can't just assume distribution of calculated property because (a)symmetries of the calculation will bite you in the ass.

Same thing that you can't just : "I don't know. Therefore 50/50."

The whole quantum thing angle doesn't make much sense to me because noone assumes there's some distribution of derived properties. Whole qm is as I understand it, calculating those distribution from underlying (a)symmetries of the model.

I would state it even more simply: an even distribution over the domain of a function is not necessarily even over it's range.

Actually, since it's a ratio, the uniform distrubution is in the logarithmic domain, ie over [−log(3),+log(3)], so the probablity (assuming, for no good reason, that it is a uniform distribution), would be:

      log 3 − log 2
  1 − ───────────── ≈ 0.81546
      log 3 − log ⅓

or just over four fifths.

But as others have pointed out, the real answer is the question is broken, since we need a distribution for x, not a interval for x.

As I understand it the key is that we do not have the distributions making it, apparently, ill-defined. If we add the Principle of Indifference the problem becomes defined enough to be solved (which is a progress) but, with two equivalent point of view, we manage to produce two different solutions (well explained in thebzax's reply). Hence the paradox.

> the real question is how was the jug filled up

This is often the case in problems of this sort: "probability" is not really well-defined until you have specified how a particular system was prepared.

Yeah, my conclusion, too. There is no paradox.