geometric probability – Distribution of line segment intersections in random pointsets

let $P$ be a set of $n$ points that are uniformly distributet inside the unit square ore unit circle, and $L=lbraceell_{ij}rbrace := lbrace lbrace alpha p+ (1-alpha q)rbrace,|,0lealphale 1;, p,qin Prbrace$ the set of line segments connecting pairs of points.

How are the numbers $operatorname{card}(lbrace ell_{hk}| lbrace h,krbracesubseteq P,setminuslbrace p,qrbracerbrace)$ of line-segments that intersect $ell_{pq}$ distributed?