given n random lines in a 2d plane, what's the probability for having less than k intersection points?

Question

In euclidean space, uniformly distributed lines (select slope and intercept uniformly from a unite square)

for example, n = 2,

when $k \leq 1$ the probability is 1. two-line at most can intersect once.

what about n = 3, 4, 5.. ?

Answer 1

There is no standard way to choose a line in the plane uniformly at random.

I suspect that with any reasonable distribution, $n$ lines chosen at random will be in general position with probability $1$. That means that with probability $1$ every line will intersect every other at just one point, with no three way intersections. That will lead to $n(n-1)/2$ points of intersection. The probability of any other number of intersections will be $0$.

You may find interesting related reading here: https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)