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A unit square is divided up with $n$ random lines. The random lines are chosen as follows, we choose one side of the square and pick a random point on that side. From there we choose a random point on one of the three other sides and connect the points. We then continue this process $n$ times.

After $n$ random lines what is the expected number of quadrilaterals formed?

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    $\begingroup$ Your title says "triangles" and your question says "quadrilaterals". Also, there is more than one way to count. Please clarify. $\endgroup$
    – Brendan McKay
    Commented Nov 3, 2017 at 6:54
  • $\begingroup$ Surely you mean polygons, not triangles or quadrilaterals? Of course you can decompose any polynomial into triangles. Is that what you mean? If this is so then a good place to start might be the number of subdivisions of the lines, since each subdivision is used exactly twice as the edge of a polygon, one on each side. $\endgroup$
    – James Smith
    Commented Nov 3, 2017 at 8:25
  • $\begingroup$ Yes, I believe I mean polygons and not quadrilaterals. Also the triangles in the title was an error. $\endgroup$
    – Pierre Humbert Leblanc
    Commented Nov 3, 2017 at 14:21

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