In this formulation, isn't p^N the probability that ALL places where life is possible, actually has life? It makes sense for that to approach zero.
What we want is the probability for at least one other place other than ours to have life. This would be 1 - (1-p)^N, which does tend to 1 as N gets arbitrarily large.
To get that formula: (1-p) is the probability that life does not exist in a place, so (1-p)^N is the probability that ALL places where life is possible, has no life. Therefore, 1-(1-p)^N is the probability of the opposite of that (where at least one place has life).
What we want is the probability for at least one other place other than ours to have life. This would be 1 - (1-p)^N, which does tend to 1 as N gets arbitrarily large.
To get that formula: (1-p) is the probability that life does not exist in a place, so (1-p)^N is the probability that ALL places where life is possible, has no life. Therefore, 1-(1-p)^N is the probability of the opposite of that (where at least one place has life).