Section 9.3

Points on a Circle

Random points on a circle: all within a semicircle or arc, triangles containing the centre or obtuse, tables that stand, arcs that overlap.

Toolkit

Place points i.i.d. uniformly on a circle of circumference $1$ ; by rotational symmetry one point can always be fixed or used as a reference origin, and the key trick is to work with arc lengths between consecutive points rather than absolute positions.

Semicircle covering: with $n$ points, $\mathbb{P}(\text{all in some semicircle}) = n\,2^{-(n-1)}$ , obtained by summing disjoint events “point $i$ is the clockwise-most point of a covering semicircle”.
Triangle contains centre: $\mathbb{P} = 1/4$ for $n=3$ (complement of the semicircle event); equivalently the centre is enclosed iff no arc between consecutive points exceeds $1/2$ .
Obtuse/acute triangle: determined by whether the arc opposite a vertex exceeds a semicircle.
For $n$ independent arcs or gaps, order statistics of uniform points split the circle into exchangeable spacings; condition on one point’s position to reduce to uniform order statistics on an interval.

Problems