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

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.. ?

• To make an issue of probability one needs to assign a probability distribution for possible "random lines in a 2d plane", or at least assume something about the likelihood that with $n$ lines, more than two will share a common point of intersection. Oct 13 '19 at 12:46
• Thanks for the reminder ! I’m interested in Uniform random Oct 13 '19 at 13:03

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 $$0$$.

• What about select ratio and intercept uniformly from a unit square ? Oct 13 '19 at 13:20
• @pengyu That is one possible distribution (assuming "ratio" is what is usually called "slope"). I conjecture that $n$ random lines will be in general position (just look at the linear equations that need to be solved). Of course many of the intersections will be outside the unit square. If you have a context in which you have reason to expect fewer intersections please edit the question to give us more of that context. Oct 13 '19 at 13:28
• thanks for correcting my language! and i'm more interested in the probability distribution instead of the expected or extreme number of intersection points. Oct 13 '19 at 14:56
• You're welcome. I think the probability distribution is boring: everything but the maximum (general position) has probability $0$. Oct 13 '19 at 21:30
• True... thanks! And sorry my context is that my brain happen to enumerate this problem for myself. by put a few concept together, and try to see if I truly understand probability. I guess learning is boring... Oct 14 '19 at 4:33