Pair of lines and maximal probability

In this paper we consider two independent and identically distributed lines, which intersect a planar convex domain $\mathbf{D}.$ We evaluate the probability $P_ {\, \mathbf{D}},$ for the lines to intersect inside $\mathbf{D}$.
Translation invariant measures generating random lines is obtained, under which $P_ {\mathbf{D}}$ achieves its maximum for a disc and a rectangle. It is also shown that for every $p$ from the interval $[0, 1/2]$ and for every square there are measures generating random lines such that $P_ {\, \mathbf{D}}=p.$