Let's say that I have a one-dimensional line of finite length 'L' that I populate with a set of 'N' random points. I was wondering if there was a simple/straightforward method (not involving long chains of conditional probabilities) of deriving the probability 'p' that the minimum distance between any pair of these points is larger than some value 'k' -i.e. if the line was an array, there would be more than 'k' slots/positions between any two point. Well that, or an expression for the mean minimum distance (MMD) for a pair of points in the set - referring to the smallest distance between any two points that can be found, not the mean minimum/shortest distance between all possible pairs of points.

I was unable to find an answer to this question after a literature search, so I was hoping someone here might have an answer or point me in the right direction with a reference. This is for recreational purposes, but maybe someone will find it interesting. If not, apologies for the spam.