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*To*: dinosaur@usc.edu*Subject*: RE: Extinction*From*: Nathan Myhrvold <nathanm@MICROSOFT.com>*Date*: Thu, 5 Dec 1996 12:00:15 -0500 (EST)*Reply-to*: nathanm@MICROSOFT.com*Sender*: owner-dinosaur@usc.edu

Killing marbles may or may not be related to dinosaurs, but the mathematics is pretty straightforward. If you have N marbles, of which X of them are some color (i.e. a species etc), then and you pick M of them without replacement, the probability that you got all X of the ones in the group is given by the hypergeometic distribution, which in this case reduces to: Prob = M! (N-X)! ------------- N! (M-X)! Where ! indicates the factorial function, and the extinction rate is M/N (i.e. the percentage you select). Factorials are large, so if you calculate this on a spreadsheet it is usually better to use the built in hypergeometric distribution function (Excel and other spreadsheets have this). Given the example discussed in this thread N = 72 X (number of dinos) = 12 M = 54 (75% extinction rate) then the probability of killing all the dinos is 2.2%. M = 61 (84.7% extinction rate) the probability of killing all dinos is 11% M = 65 (90.3% extinction rate) it rises to 26.2% The obvious problem with this model is that it assumes that species are killed indepedently - this is certainly NOT the case for predators which depend on a prey species, or for a set of species that depend on a common resource (i.e. if a lake dries up ALL the fish die, not some independent sample of them). One crude way to estimate this is to assert that if any 11 of them went extinct, the remaining one would go too because it is not viable, or would be correlated. This dramatically changes the odds. 75% extinction M = 54, probability of 11 or more species going extinct = 13.5% 84.7% extinction M = 61, probability of 11 or more species going extinct = 41.2% 90.3% extinction M = 65, probability of 11 or more species going extinct = 67% Note that this approach is based at the species level. However, you could just as easily think of the marbles as individuals, and ask what level of killing the population would leave a remainder that wasn't viable. I strongly doubt that the K/T extinction should be modeled as purely independent events, like picking marbles. Some correlation was there. The interesting thing about simple probabilistic models is that you can gain some interesting insight and constraints on the problem. Nathan

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