For a random variable X taking on non-negative integer values (here, the number of occurrences of life elsewhere in the universe), by Markov's inequality the probability that X = 0 is >= 1 - E[X]. Here, E[X] = Np, so if Np is very close to 0, the probability that X = 0 will be very close to 1.

That the probability goes to 1 as N goes to infinity FOR FIXED p is just another example of assuming p can't be "too small". The probability also goes to zero as p goes to zero. Why are you fixing p and not N? Why are you assuming p is large enough that N is in that asymptotic range where the probability has approached 1?

That seems right, but from a scientific point of view (as opposed to, say, a certain sort of theological view), two occurrences is not much more than one (even though one is so much more than zero.)

Two occurrences would actually be much more than one! Our own existence is useless due to observer selection, but discovery of even a single other independent OoL event nearby would allow us to infer OoL cannot be too uncommon.

Observer selection does not eliminate us as evidence for the proposition that life can exist. As for whether it is rare, you added the qualification 'nearby', and while it is true that it is most likely that any extraterrestrial life we detect will be nearby, the post I was replying to was arguing about the universal probability of life coming into existence, not about whether it will be discovered by us.

Furthermore, proponents of an extraterrestrial origin of life on Earth will doubtless argue that nearby life may have had a common origin.

Observer selection means p > 0 (ie the inequality is strict) but it can't tell us any more. Bayesian reasoning from our own solar system can put a reasonable upper limit on p but that isn't very helpful.

However, if we found life on Mars that same Bayesian reasoning would imply a meaningful lower limit on p as well, since life on Mars is independent of our existence to observe it.

If we found life on Mars that was independent of life on Earth it would imply a meaningful lower bound. Even finding a fundamentally different biosystem on Earth (life that didn't use nucleic acids, say) would be informative.

Just finding life on Mars that's the same kind of life as on Earth would not tell us much, as it could be explained by panspermia. There are Mars rocks on Earth, so transfer of life in those rocks should have happened constantly. If early Mars were habitable it almost certainly had life, due to this transfer.

> However, if we found life on Mars that same Bayesian reasoning would imply a meaningful lower limit on p as well.

If we found life on Mars tomorrow, how well-defined would that lower limit become?