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I was incredulous that this was as simple as I first thought[1], but had my suspicions later confirmed when I read about it on lesswrong [1] http://iaindooley.

by dools 11y ago

I was incredulous that this was as simple as I first thought[1], but had my suspicions later confirmed when I read about it on lesswrong [1] http://iaindooley.com/post/61292324101/the-two-envelopes-problem-the-most-boring-paradox http://iaindooley.com/post/61292324101/the-two-envelopes-pro...

dllthomas 11y ago
I don't think it's as simple as you thought. Or at least, the problem you identified is not a problem. The actual problem is still pretty simple. [for those dropping into this thread without having followed the link, we seem to have restricted our discussion to the (x, 2x) problem, which is mentioned in the plover.com article but is not where it starts] Probability is a means of dealing with uncertainty. When you open the envelope you were handed and see $10, what you've learned is that the other envelope contains either $5 or $20. This is not postulating existence of both of those envelopes - this is your uncertainty about which world you inhabit, one with two envelopes ($5, $10) or one with two envelopes ($10, $20). It's true that you only inhabit one of those worlds, but you don't know which and probability is how you get a handle on your lack of knowledge. As I understand it, the problem is that a natural - but errant! - next step is to say, "I don't know how likely it is that the envelopes were stuffed with $5 and $10 vs $10 and $20, so I'm going to assume both are equally likely." Around $10, this is probably a good approximation. But if you're saying this before looking at the envelopes, then it amounts to saying "the probabilities are equal for any choice of value" (for all x, P(x)=P(2x)). I'm actually not sure what distribution should be chosen to reflect zero knowledge of a rational number, but it should be a distribution - the likelihood of all possible outcomes should sum to certainty (1). That is incompatible with "for all x, P(x)=P(2x)".

by dools 11y ago

dllthomas 11y ago

I don't think it's as simple as you thought. Or at least, the problem you identified is not a problem. The actual problem is still pretty simple. [for those dropping into this thread without having followed the link, we seem to have restricted our discussion to the (x, 2x) problem, which is mentioned in the plover.com article but is not where it starts] Probability is a means of dealing with uncertainty. When you open the envelope you were handed and see $10, what you've learned is that the other envelope contains either $5 or $20. This is not postulating existence of both of those envelopes - this is your uncertainty about which world you inhabit, one with two envelopes ($5, $10) or one with two envelopes ($10, $20). It's true that you only inhabit one of those worlds, but you don't know which and probability is how you get a handle on your lack of knowledge. As I understand it, the problem is that a natural - but errant! - next step is to say, "I don't know how likely it is that the envelopes were stuffed with $5 and $10 vs $10 and $20, so I'm going to assume both are equally likely." Around $10, this is probably a good approximation. But if you're saying this before looking at the envelopes, then it amounts to saying "the probabilities are equal for any choice of value" (for all x, P(x)=P(2x)). I'm actually not sure what distribution should be chosen to reflect zero knowledge of a rational number, but it should be a distribution - the likelihood of all possible outcomes should sum to certainty (1). That is incompatible with "for all x, P(x)=P(2x)".