Let us attempt engagements with the thrust of OP's problem. The following thoughts are also offered without proof by way of sketch, and deserve checking.
1) Say that the space is R, the real number line (which serves for a true line, in the literal mathematical sense, extending infinitely in both directions). L meanwhile is some finite radius (open, half open, closed? I don't see how it's material at this trivial case). P1 is generated anywhere on the real line, RANDOMLY, excluding its radius. Can't do P2 anywhere inside P1's L radius. We have an iterative process. now do P2, RANDOMLY. same thing, same L.
Clearly infinitely many P's may be generated in a a straightforward iterative fashion (is the axiom of choice tacit in this?), in an irregular way. OP's question is thus still valid in this version and I don't know how to go about it, here.
2) Consider instead of the line R, the ray [0,inf) , or the appropriate caveat(?) (0,inf). We still have infinitely much space to work with, generating infinitely many P's per the above treatment. Again, I don't know how to approach this.
3) Now the interesting part. Consider some finite line-segment-tier subset of R, being either closed, open, or half-open intervals as subcases and interest require, now apply the above P-L logic. Having only finite space to work with, (with respect to some L), it seems reasonable that the space should be finitely exhausted, in a general way. Thus OP's thing of "very long line" as opposed to "infinite line" proves to be germane after some thought. Some boilerplate about intervals, points, etc needs to be pinned down I would suggest, along with the statistical stuff that I admitted to not knowing.
I invite others to develop this train of thought further.