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Take the area $[0,1]^2$ which is intersected by $n$ random lines. The way we can choose these $n$ random lines is by choosing two points on the border of the square and connecting them.

If we do this, what is the probability that a region formed by these lines has area greater than $1/2$. I can see that for $n=1$, the probability is equal to one.

I could probably approximate the value through a computer simulation of some sort but I'm just wondering if anyone can find a pure math answer.

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    Are the points on the border of the square chosen independently, uniformly at random from the perimeter? What happens if one pair of points is chosen from the same side of the square—do we try again, or does that ineffective line count? – Greg Martin Jun 21 '17 at 00:14
  • Also, what is the "region formed by the lines"? Three lines will clearly define a triangle (though this may extend outside the square), but if we add a fourth, it will either cut the triangle or lie entirely outside it. In either case, what is the shape of the region? – Graham Kemp Jun 21 '17 at 00:22
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    @Graham I interpreted "a region" to mean "any region" i.e. the largest. Definitely needs some OP clarification though. – Quickdraw Jun 21 '17 at 00:29
  • Yes, a region means any region of any shape. Yes, the points are chosen independently. If you choose a point which is on the same side of the square you disregard that line. –  Jun 21 '17 at 00:46
  • Also, just noting any ideas or tips would help too, please post it if you have anything... –  Jun 21 '17 at 00:47

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