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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 incomplet

by bhk 8y ago

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.

sjg007 8y ago
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.
- bhk 8y ago
  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.

by bhk 8y ago

sjg007 8y ago

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.

bhk 8y ago
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.