In the solution, it is stated that:

"The probability that at least one of them will have an infinite family tree is 1 - (1 - P)^2, because (1 - P)^2 is the probability that both of them will perish undivided."

But this seems to simplistic to me. The implicit claim here is that 1 - P (which is the probability that the family tree will not be infinite) is equal to the probability that the amoeba in question will die out. But this is not the case. For example, if the amoeba does not die out, it is still possible that both its offspring will die, thus ending its chances of an infinite family tree.

Am I right or am I missing something?

I don't know how to go about it mathematically, but my intuition (which is notoriously wrong in probability questions) tells me that the probability of an infinite family tree is zero, since certainly there is a chance (albeit a very small chance) that ALL of the amoebae will die out all at once, no matter how many of them there are at any given time. And given infinite time, this is going to happen eventually.

Am I right or am I missing something?

Thanks,
ben

If by "die out" you mean dying its individual death without producing offspring, no. It only means that the family tree is finite.

>But this is not the case.

I agree.

>For example, if the amoeba does not die out, it is still
>possible that both its offspring will die, thus ending its
>chances of an infinite family tree.

I agree with that too. If P is the probability of having an infinite tree, 1 - P is the probability of having a finite tree, not just a root without a trunk.

>Am I right or am I missing something?

I believe you are. Have I explained anything? I meant to note that P accounts for the generations to come out of a single individual.

>I don't know how to go about it mathematically, but my
>intuition (which is notoriously wrong in probability
>questions) tells me that the probability of an infinite
>family tree is zero, since certainly there is a chance
>(albeit a very small chance) that ALL of the amoebae will
>die out all at once, no matter how many of them there are at
>any given time. And given infinite time, this is going to
>happen eventually.

Not 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's egg's survival is one in millions - or, in any event, something grotesquely small. But look at them. The species have been around for a few million years.

So, I think having a multiplication rate of 3/4 gives the amoeba a real chance to meet the end of the world.

On the other hand, in "real world" 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.

>
>Am I right or am I missing something?
>
>Thanks,
>ben