[AaronBrown]

I think we each have stated our positions as clearly as we can, and still think the other misunderstands us. Not much point in going forward, but I will pick out a few sentences that demonstrate our disagreement.

You seem to treat this as a paradox designed by people who have envelopes and don't know whether or not to switch. It is in fact a paradox designed to demonstrate that common statistical reasoning can lead to contradictory conclusions.

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I still don't know what you mean by "consistent prior beliefs about everything".

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Bayesian statistics defines probability as subjective belief. It assumes you can always determine a prior belief before the experiment (in this case, before the envelope is opened). The information in the experiment (in this case, the amount in the envelope) adjusts that belief to the posterior distribution. To a Bayesian, showing that no consistent prior distribution justifies switching regardless of the amount you observe, shows that always switching is irrational. My objection is that in real decisions, people often don't have consistent prior distributions.

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I don't see how the observation that people often do bad math relates to the envelope paradox.

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It's not bad math not to have a consistent prior distribution. It's impossible to have consistent beliefs about everything. It's not bad math to compute expected value in different units from someone else. Everyone has different units if you look carefully enough.

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There's no need to assume (b). (b) is simply a true statement. Also, if by (a) you mean that if you chose one of the two closed envelopes there's a 50-50 chance it will have the larger amount then that is simply a true statement as well.

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You insist both statements are true and regard them as not only beyond argument, they are so obvious, they are beyond the need to state as assumptions. The paradox was designed to teach you they cannot be both true all the time.

There are senses in which each of them are true, but they are different senses.

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if you chose one of the two closed envelopes there's a 50-50 chance it will have the larger amount then there is a repeatable experiment you have in mind that can prove out this statement of probabilty. The experiment is for numerous people to pick one of the two envelopes at random. The more people who pick, the closer you will see half picking the higher amount and half the lower amount. There's a 50-50 chance that any one of them will pick the higher amount.

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This is the frequentist argument. If that's your definition of probability, then (b) is not always true. This is the criticism that led to Bayesians to reject that definition of probability. Bayesians achieve consistency by rejecting (a) instead.

I understand the desire to make both (a) and (b) always true, but no one has discovered a way to do it.

When you have nothing in one of the envelopes, (b) no longer applies. It only covers cases of $1 and $2, not $1 and $0. You can't reconstruct the paradox in this case.