Yeah, I think Unreality is right about it not being a simple 1-(1/4)^n geometric series...the answer should be an infinite series of conditional probabilities.
[spoiler='Conditional Probability
']Like, you can't just sum the P(ext), as you call them, of the ith generations. You need to sum the P(exti)P(survi-1), where P(survi-1) is the probability of survival to the i-1 th generation. Put in less mathematical terms, it means you need to weigh the probability of extinction of that generation with the probability that the amoeba will actually survive to that generation. So it should be something like P(survival to infinity) = 1-sum(P(exti)P(survi-1)), and in the P(survi-1), you need to take into account the different cases where 1 out of 2 survives or both survive...
I haven't done the math yet, but it wouldn't surprise me it that series came out to sum to 2/3 as well...once I have time, I'll try to figure out the exact mathematical expâ€‹ression for the series