Sylvester's four point problem

Sylvester's four point problem in geometric probability asks for the probability that four randomly chosen points in the Euclidean plane form a convex quadrilateral. Together with Buffon's needle problem, it has been called "one of the prime paradigms in geometric probability theory".

The answer depends on the probability distribution from which the points are drawn, and finding a distribution for which this probability is small is closely connected to the crossing number of complete graphs. The problem was posed in 1864 by J. J. Sylvester, who asserted (with fallacious reasoning) that if the points are drawn from the entire plane then the probability is ${\tfrac {3}{4}}$ but that if they are drawn from a bounded convex set then the probability is bounded below ${\tfrac {3}{4}}$. (Sylvester originally posed the problem in a complementary form, asking for the probability that four points do not form a convex quadrilateral, and giving answers ${\tfrac {1}{4}}$ and bounded above ${\tfrac {1}{4}}$.)

Among continuous uniform distributions over bounded convex sets the probability of a convex quadrilateral is maximized by any circle or ellipse (probability approximately 0.704) and minimized by any triangle (approximately 0.667). For uniform distributions over bounded open sets the minimum probability that can be achieved is unknown, but has upper and lower bounds that are both near 0.380.