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Originally Posted by Ken G
You have used a different set of rules to derive the "snarkophilus anti-catastrophe", namely, that humanity will never become extinct, because each total number of humans greater than our present birth number is equally likely, and there are a virtually infinite number of possibilities! Interesting. This seems as justifiable as the Carter catastrophe if yours are in fact the right rules, but really neither set of rules are very plausible, just as the "everything has a 50/50 chance, it either happens or it doesn't" is also not a very useful rule. What do you say to the alien time traveller argument, in relation to your picture of the right rules?
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I'm not entirely sure what you mean by saying that there are different sets of rules. That idea came up in response to
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Originally Posted by snarkophilus
"Saying you perform the experiment and then check the value is not the same as fixing the value and then performing the experiment!"
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The question posed by Carter is, "given that you fix the result of the experiment beforehand, then perform the experiment and discover that the answer is 7, what can you say about which urn was selected?"
My answer is "nothing," because the experiment is fixed. The probabilities of having chosen each urn are the same as they were before the experiment. (I arbitrarily decided that there was a 50/50 chance of picking a particular urn. It doesn't have to be that way. That's partly why you get the "anti-catastrophe" scenario. Choosing a different distribution of urns gives a different set of probabilities.)
I suppose that another set of rules is not fixing the experiment, but performing it and seeing what happens. I've already explained what conclusion you can draw from that experiment, what you call the anti-catastrophe. And, as I mentioned, that conclusion isn't really valid in real life because there's no reason to think that each urn carries equal weight. But it's still better than the other version, because it is based upon sound logic.
To resolve the anti-catastrophe, do the following: pick a number. An integer. Any positive integer. Now, amongst the integers, the probability that you chose that number is essentially 0. (There is a discussion about surreal numbers kicking around here somewhere that is almost pertinent...) The fact is, however, that you chose an integer. That integer is finite, even though there are an infinite number of possible choices. And so it is with the anti-catastrophe. Even though you have an infinite number of choices of urns, in the end you have to pick one, and it has a finite number of balls in it.
There's still the possibility that the anti-catastrophe occurs, of course, but you can't decide how probable that is compared to an integral count (catastrophe after n people) with additional information.
*pause for breath*
And that leads in to the alien time traveller.
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Originally Posted by Ken G
The time traveller tells you that he has observed the extinction of a million intelligent species on a million worlds. He amuses himself by picking lives completely at random, and paying them a visit, and he chose you. And he mentions, by the way, that of course by these rules, 90% of the time he is talking to someone whose species does not outnumber that individual's birth order by more than a factor of 10.
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He has chosen a life at random from the species, so you have a 90% chance of being one of the last 90% of people to be alive. That makes perfect sense, and doesn't worry me in the slightest. That's because you have to add up all the possibilities. If there are only ever going to be 10 billion people, and I'm number 10 billion, that's bad. Same for if there are only going to be 11 billion. But what about if there are going to be a trillion? Two trillion? There are a lot of numbers up there, and in each of those scenarios, you're in the first 10%. It's increasingly improbable that he would choose you in particular as species number increases, but the fact is that he had to choose
someone. In fact, if there's truly an equal chance of your species number being any number, your birth number is statistically so small (compared to infinity) that there's no point in worrying: you're most likely at 0%.
You need a realistic distribution as to the probabilities of each species number being correct to get any information out of your extraterrestrial visitor.
And, of course, there's the possibility that you'll never be extinct. He's witnessed a lot of extinctions, sure. But he didn't say how many species go on forever. You have no data whatsoever about that (except the laws of thermodynamics, but I digress).