Abstract
In this article we provide several exact formulae to calculate the probability that a random triangle chosen within a planar region (any Lebesgue measurable set of finite measure) contains a given fixed point O. These formulae are in terms of one integration of an appropriate function, with respect to a density function which depends of the point O. The formulae provide another way to approach the Sylvester’s four-point problem. A stability result is derived for the probability. We recover the known probability in the case of an equilateral triangle and its center of mass: \(\frac{2}{27}+20\frac{\ln 2}{81}\) (Halasz and Kleitman in Stud Appl Math 53:225–237, 1974; Prekopa in Period Math Hung 2:259–282, 1972). We compute this probability in the case of a regular polygon and its center of mass for the point O. Other families of regions are studied. For the family of Limacons \(r=a+\cos t, a>1\), and O the origin of the polar coordinates, the probability is \(\frac{1}{4}-\frac{12a^2(4a^2+1)}{(2a^2+1)^3\pi ^2}\).
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