In this paper we consider two independent and identically distributed lines, which intersect a planar convex domain D. We evaluate the probability P, for the lines to intersect inside D. Translation invariant measures generating random lines is obtained, under which P 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=p.

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