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 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?