[AaronBrown]

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I would very much like to see you write this down rigorously -- i.e., formally define your probability space, your probability measure(s), and your random variables. You claim to be demonstrating fundamental flaws in probability theory that stem from the Bayesian/Frequentist debate. But without a rigorous presentation, I fail to see how you've demonstrated anything other than (a) the uniform distribution fallacy, or (b) the fact that being inconsistent about what is random and what is fixed, and about what is conditional and what is not, can lead to the wrong answer.

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To answer your specific question, X is a variable, but not a random one. It is the amount in one envelope. The amount in the other envelope is a random variable, with expectation 1.25*X.

You have inverted my meaning. I claim the envelope paradox is a true paradox, two strong arguments that lead to opposite conclusions. Yes, people invent rigorous ways to resolve it, but only by torturing commonsense ideas of uncertainty. They end up with mathematically consistent theories that give absurd results in some cases.

I don't dispute the rigor of either the Bayesain or Frequentist approaches, merely their relevance to practical decision making. The Bayesian says the odds are not 50/50, the Frequentist says the expected value of a 50% chance of getting 0.5*X and a 50% chance of getting 2*X is not 1.25*X. Both say you shouldn't always switch, which agrees with common sense, but only by invoking complex technicalities. In practical decision making, you'd never see these technicalities, so you'd fall into error. Moreover, there are situations in which it does always make sense to switch, the technicalities give no clue as to when those situations occur.

I'm not anti-statistician, I just think people internalize the mathematical argument and treat it as necessary for reality. It's useful in some cases, not in others. It's good for predicting the chance of getting dealt pocket Aces, bad for telling you when to call an all-in bet on the river.

I don't have a rigorous framework, if I did I'd publish and get the Nobel prize. My claim is no one can give clear, commonsense answers to both of the following questions without leading to absurd conclusions:

(1) Is the probability that you have the higher amount 50%?

(2) If you have a 50% chance of losing some amount and a 50% chance of winning twice that amount, is it a positive expected value bet?

Imagine you have an envelope in your hand, with $10,000 in it. You wonder whether or not to switch. The Bayesian tells you to look inside yourself to figure out what probability you assigned to every possible total amount in the two envelopes before you opened yours. Doesn't that sound a little bit unhelpful? Your only question now is how likely is a $15,000 total versus a $30,000 total? That's what you should be weighing, using your current state of mind, not your prior one.

The frequentist tells you that based on the design of the experiment, he believes that if it were repeated many times, you would end up with the higher amount 50% of the time, therefore, the probability you have the greater amount is 50%. You then say, "Great, you're telling me that the expected value of switching is $12,500." "No," he replies, "that computation is disallowed." Aren't you going to want to take a swing at him?