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matt

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 [Oct. 30th, 2007|03:52 pm] matt The shooting-room paradox was introduced by John Leslie (e.g. [1996], pp. 251ff), who says he developed the idea with help from David Lewis, who considers it "a good, hard paradox".In the shooting room experiment we are to imagine a room of infinite capacity. First a batch of ten people are led into this room. A pair of dice is thrown in front of their eyes. If a double six comes up they are all shot. Otherwise they leave the room safely and a new batch, this one containing a hundred people, is thrust in. The process continues, with each consecutive batch ten times larger than the previous one, until there is a double six; whereupon the people in the room at that time are shot and the experiment ends.Suppose you have been thrust into the room. You are asked to estimate the odds of leaving safely. One the one hand, since whether you will leave or not will be determined by the throw of a fair pair of dice, it seems that you have a 35/36 chance of exiting alive. On the other hand, 90% of all people who are in your situation will be shot, so it seems you have only a 10% chance of exiting alive. That is the paradox.-- from http://anthropic-principle.com/preprints/lit/index.html via http://sciencesecurity.livejournal.com/61385.html Link Reply

 From: 2007-10-30 11:18 pm (UTC) (Link)
of course, if you have just been thrust into the room, it is also true that 100% of the people who have been in your situation have exited alive.
 From: 2007-10-30 11:54 pm (UTC) (Link)
I think that's Leslie's conclusion: if I continue to observe the experiment then doomsday has not happened, so the probability is 0%, anthropic principle QED.

 From: 2007-10-31 04:39 am (UTC) (Link)
This sounds to me like the two-envelope paradox. You're given an envelope and then told that there's a second envelope and that one ontains twice as much money as the other. Should you pay a nominal amount to switch? If so, should you now pay to switch back, before opening either? Etc. I think these are paradoxes of infinity - the supply (of money, of victims) is inexhaustible and so the contradiction is pushed out towards infinity. With the shooting room the question is whether you find yourself in the experiment or not. 91 per cent of the people in the experiment die, but there's only a 2.9 per cent chance you'll be in that 91 per cent because the pool keeps expanding and there's no limit to its expansion. If there were a limited pool either it would be large and they'd be unlikely to reach you, or it would be small and they'd be unlikely to kill 91 per cent of people because there just wouldn't be the expected 10 to the 35th people available. Or to put it another way, each iteration is a new experiment with a different and expanding set of participants. When it's your turn you're probably not participating in the100 per cent dead experiment. Very interesting - thanks for posting it.
 From: 2007-10-31 05:21 am (UTC) (Link)
There's no paradox. The puzzle is set up to hide the fact that the probabilities are set up over two different populations. Your survival chance is 35/36 if you are inside the room at a particular moment, 100% if you've been inside and are already out, and dependent on the size of the total population if you haven't been picked to go inside yet.

One can go further; if the pool of potential victims is finite, there's a nonzero chance that everyone will escape alive. If the pool of potential victims is infinite, probability theory doesn't apply.
 From: 2007-10-31 05:29 am (UTC) (Link)
Skimmed the preprint. Yeah, that's not a paradox, just bloody-minded wooly thinking. Philosophers seem to get free papers when they get math wrong. Lucky bastards.
 From: 2007-10-31 05:47 am (UTC) (Link)
Actually the more I think about it, the angrier I get. The "paradox" isn't just wrong, it's howlingly wrong and could be cleared up by talking to someone who actually knows something about probability for five minutes. But then they wouldn't get a paper out of it. GRRRRRR.
 From: 2007-10-31 01:53 pm (UTC) (Link)
yup. of late, i've read too many papers in the "soft sciences" that are no better reasoned (and often worse reasoned) and i've lost what little confidence i had in the fields.
 From: 2007-10-31 03:11 pm (UTC) (Link)
In general, I agree that there's no paradox. But, what worries me is the last statement you're making there: there's never any question that there's a nonzero chance that everyone will escape alive: as I see it, at any given instant, there should be a 97.2% chance that everyone will escape alive, and that past experiments don't matter to the individual. The house always wins, though.

However, I'm still bothered by all that. To my thinking, infinite timeframes shouldn't matter from a calculational standpoint. Of course, I should probably sit down and think out if this is a converging series before I say that, but I have similar difficulties with questions like Schrödinger's cat. For example, let's say it's a pair of fecund cats. As time progresses, the number of cats increases exponentially, but the chance that all the cats die is also increasing. It seems reasonable that the mean number of cat corpses in the box might be computable, if I'm following the experiment from beginning to end.

That said, if I'm randomly sampling a known experiment which I have no idea when it began, I can't make conclusions like John Leslie is making. It's the guessing, that I'm near the "end" or near the "beginning" that really bothers me.