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jkaufman comments on Probability puzzle: Coins in envelopes - Less Wrong Discussion
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Comments (18)
I must be missing something: both 1 and 2 appear to be impossible. He picked a random envelope, it had a coin, he removed it, she put that envelope, now empty, back with the others. How can the expected value of picking an envelope not have decreased? There are the same number of envelopes, and one fewer coin.
He also finds out that the envelope wasn't empty. If the envelopes are positively correlated -- knowing that one envelope has money increases the probability that another envelope has money -- then this can counteract the effect of replacing this envelope. The trick is to make the correlations be such that the two effects cancel out exactly.
Yeah, that's the solution. And one that should be obvious to anyone familiar with Jaynes. Probabilities are about states of knowledge, not about physical propensity.
Unfortunately, although I'm familiar with Jaynes, I jumped to a propensity interpretation. Fewer coins must mean a lower chance of picking a coin - which it obviously would, for someone whose estimate of the total number of coins doesn't change. And then I spent my energy marshaling the arguments why these wackos must be wrong and don't know diddley.
My takeaway is that it is usually more useful to ask - how something could be true, than why it must be false. I think the solution would have been obvious if I spent my energy looking for it, instead of denying it's existence.
Also, I should takes pains to keep in mind that the other guy isn't a moron, and when I limit myself to 5 seconds of honest reflection on a problem, I am.
That's why I expected to "get" fewer people on LW than on xkcd. One welcome surprise was that it seems to have served as an intuition pump for one person over there, who had, only a few days earlier, written
This same person initially responded that the problem was impossible, but then was enlightened:
I'm having trouble understanding how you (and buybuydandavis) see this puzzle as illustrating (or evidencing?) a subjective approach to probability. Wouldn't it be perfectly solvable in the frequency/propensity approaches in just the same way? Conditional probability and the Bayes rule work the same way everywhere.
(I haven't read Jaynes yet) (Also enjoyed working out your puzzle, and reposted it in my blog, hope you don't mind)
Certainly don't mind. It's certainly solvable with a propensity approach, it's just that the problem description points you toward the wrong kind of propensity: there really is an absolute proportion of coins to envelopes that has strictly decreased, but that's not the relevant value.