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Originally Posted by Ken G
Let me clarify-- first N is selected by the gamemaster, and that selection is what has an unknown distribution for us. After N is chosen, then we get a random number from 1 to N. The latter is evenly distributed, so the whole game exactly mimics the Carter situation, but is more conducive to mathematical scrutiny.
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Yes, but to then add the aspect of only looking at pre-selected values of M no longer matches the Carter situation.
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There is no "strength" to the Carter hypothesis. It can only be one of two things depending on how far you take it: 1) the plainly obvious statement that 90% of beings live in the last 90% of any set from which they are generically chosen, and 2) the false probability argument that this says anything about the number of humans that will be born, given the number that have been. So the problem is not in the "random sampling", it is in the incorrect use of probability concepts-- and the mathematical game I described shows this, just take any interesting distribution over N that you like and ask if 50% of the times that a given specified M is chosen, will that M be > N/2. It will not-- unless you average over M, but then you can't use M to create the whole catastrophe concept.
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The Carter hypothesis does not select a given "M" and so this argument does not apply.
The fact that "the plainly obvious statement that 90% of beings live in the last 90% of any set from which they are generically chosen" is the power of the Carter hypothesis when simply combined with an exponential population curve and assumption that all known species have a finite existence. That is the strength of the argument - that a statistical estimate can be made with so little input. It is also the weakness of the argument because it requires one to ignore any additional knowledge.
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It is a common misconception about probability that it has something to do with your knowledge. The choice to include knowledge is yours when you do a probability calculation, there is nothing "automatic" about it. All probability calculations are subject to the assumptions you put in-- knowledge is irrelevant, except that it is normally assumed that you will use all your knowledge in building your assumptions. When people think that probabilities depend on knowledge, they get all confused about under what situations do the probabilities change, like if you forget what cards have been shown in poker, does your probability of winning change? Answer: there is no unique concept of a probability of winning, it all depends on the calculation you choose to make.
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Well, it certainly is important to include knowledge into an probability calculation if such knowledge is available to improve the validity and reliability of the calculation. Certainly, one's calculation is only as good as the assumptions that went into the calculation. If knowledge is available but not used, then any such calculation is limited by the assumptions that were used (are the dice loaded?)
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It still applies, if you choose not to use any argument or evidence-- on the grounds that it would be of suspect reliability (which it would). The problem with Carter is much more fundamental-- it is wrong probability if you are selecting on the basis of the value of M.
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If you chose not to use any other information to help describe the likely population distribution (and have a value of M as a result of an essentially random selection {not-pre-selected} from a finite population), then the Carter estimate is the best you can do - but it is fatally weak because you are purposely not using important information that may have significant impact on the likely population curve and expected specie longevity.
(It is like expecting the normal odds of getting a 7 is a valid statistical argument even when you know that the dice are loaded but you just choose to ignore that information and calculate odds assuming that you don't know they are loaded).