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> Given a bounded rectangular area, I generate 4 random points. What is the probability that the fourth point lie within a triangle formed the first 3?
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How would I attack this problem? The goal is to eliminate points by knowing whether they are encompassed by any 3 other points. 



On 4 random points in a rectangle
As explained in [Square Triangle Picking](http://mathworld.wolfram.com/SquareTrianglePicking.html), the mean area of a triangle picked inside a rectangle of unit area is 11/144. So the probability that the fourth point lands inside this triangle is $11/144=0.0764$.


See [Polygon Triangle Picking](http://mathworld.wolfram.com/PolygonTrianglePicking.html) for the corresponding probability $p\_n$ when the square is replaced by any regular $n$-polygon,
$$p\_n=\frac{9\cos^2(2\pi/n)+52\cos(2\pi/n)+44}{36n^2\sin^2(2\pi/n)},$$
a remarkable formula derived by Alikoski in 1939. The $n\rightarrow\infty$ limit gives the probability $35/48\pi^2=0.0739$ for a circle (equal to the average area of a triangle inside a disc of unit area).


