Four Points in a Disk
Four points are chosen independently and uniformly at random inside a disk. What is the probability that one of the four points lies inside the triangle formed by the other three (so that their convex hull is a triangle rather than a quadrilateral)? Give an exact closed form, or a decimal rounded to 4 places.
Related problems & prerequisites
Worth solving first
- One Point per Side 3.0/10
- Ten Letters, Ten Envelopes 3.0/10
- Monochromatic Triangles 2.0/10
Source: Sylvester's four-point problem (J. J. Sylvester, 1864), here for a disk; the value is classical. Statement written for AxiomIQ.