X Marks The Spot

As illustrated in the diagram, you draw two random line segments on a circular piece of paper.

What is the probability that the two segments intersect?


Bonus: Generalize the result for an n n -sided regular polygon paper.

1 3 15 16 π 2 \dfrac{1}{3}-\dfrac{15}{16\pi^2} 1 3 35 36 π 2 \dfrac{1}{3}-\dfrac{35}{36\pi^2} 2 3 25 26 π 2 \dfrac{2}{3}-\dfrac{25}{26\pi^2} 2 3 5 6 π 2 \dfrac{2}{3}-\dfrac{5}{6\pi^2}

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