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0|0|0|||||Amoeba%27s survival problem.|kakatooie||14:27:43|02/21/2008|This is in regard to the Amoeba%27s survival problem%2C which asks%3A%0D%0AGiven one amoeba%2C who has a 3%2F4 chance to split into two identical amoebae%2C and a 1%2F4 chance of dying out%2C what is the probability P that its family tree will be infinite%3F%0D%0A%0D%0AIn the solution%2C it is stated that%3A%0D%0A%0D%0A%22The probability that at least one of them will have an infinite family tree is 1 - %281 - P%29%5E2%2C because %281 - P%29%5E2 is the probability that both of them will perish undivided.%22%0D%0A%0D%0ABut this seems to simplistic to me.  The implicit claim here is that 1 - P %28which is the probability that the family tree will not be infinite%29 is equal to the probability that the amoeba in question will die out.  But this is not the case.  For example%2C if the amoeba does not die out%2C it is still possible that both its offspring will die%2C thus ending its chances of an infinite family tree.%0D%0A%0D%0AAm I right or am I missing something%3F%0D%0A%0D%0AI don%27t know how to go about it mathematically%2C but my intuition %28which is notoriously wrong in probability questions%29 tells me that the probability of an infinite family tree is zero%2C since certainly there is a chance %28albeit a very small chance%29 that ALL of the amoebae will die out all at once%2C no matter how many of them there are at any given time.  And given infinite time%2C this is going to happen eventually.%0D%0A%0D%0AAm I right or am I missing something%3F%0D%0A%0D%0AThanks%2C%0D%0Aben
1|1|0|||||RE%3A Amoeba%27s survival problem.|alexb||15:28:43|02/21/2008|%3EBut this seems to simplistic to me.  The implicit claim here %0D%0A%3Eis that 1 - P %28which is the probability that the family tree %0D%0A%3Ewill not be infinite%29 is equal to the probability that the %0D%0A%3Eamoeba in question will die out. %0D%0A%0D%0AIf by %22die out%22 you mean dying its individual death without producing offspring%2C no. It only means that the family tree is finite.%0D%0A%0D%0A%3EBut this is not the case.  %0D%0A%0D%0AI agree.%0D%0A%0D%0A%3EFor example%2C if the amoeba does not die out%2C it is still %0D%0A%3Epossible that both its offspring will die%2C thus ending its %0D%0A%3Echances of an infinite family tree. %0D%0A%0D%0AI agree with that too. If P is the probability of having an infinite tree%2C 1 - P is the probability of having a finite tree%2C not just a root without a trunk.%0D%0A%0D%0A%3EAm I right or am I missing something%3F %0D%0A%0D%0AI believe you are. Have I explained anything%3F I meant to note that P accounts for the generations to come out of a single individual.%0D%0A%0D%0A%3EI don%27t know how to go about it mathematically%2C but my %0D%0A%3Eintuition %28which is notoriously wrong in probability %0D%0A%3Equestions%29 tells me that the probability of an infinite %0D%0A%3Efamily tree is zero%2C since certainly there is a chance %0D%0A%3E%28albeit a very small chance%29 that ALL of the amoebae will %0D%0A%3Edie out all at once%2C no matter how many of them there are at %0D%0A%3Eany given time.  And given infinite time%2C this is going to %0D%0A%3Ehappen eventually. %0D%0A%0D%0ANot long ago the PBS showed a piece about the horseshoe crab and the red knot migration. Numerous birds feed on horseshoe eggs and the hatchlings and in fact depend on that food for their survival. The bottom line  is that the chances of a horseshoe crab%27s egg%27s survival is one in millions - or%2C in any event%2C something grotesquely small. But look at them. The species have been around for a few million years.%0D%0A%0D%0ASo%2C I think having a multiplication rate of 3%2F4 gives the amoeba a real chance to meet the end of the world. %0D%0A%0D%0AOn the other hand%2C in %22real world%22 one oil spill next to their spawning beaches has a chance of ending abruptly a million year dynasty. The problem does not take into account the real world vagaries. Certainly.%0D%0A%0D%0A%0D%0A%3E%0D%0A%3EAm I right or am I missing something%3F %0D%0A%3E%0D%0A%3EThanks%2C %0D%0A%3Eben %0D%0A