Section 9.4

Discs, Spheres and Darts

Uniform points in discs, on spheres or on square boundaries; dartboards and targets; Buffon-type problems.

Toolkit

Uniform points in a planar region, on a surface, or on a sphere have probability of lying in a sub-region equal to the ratio of measures (area, volume, or surface area) to the total. For uniform directions in $\mathbb{R}^3$ , Archimedes’ hat-box theorem says the surface area of a spherical zone between two parallel planes is proportional to the distance between the planes, not to the shape of the cut, so probabilities for direction-based problems reduce to a height ratio.

Key tools:

Centre-restriction: a shape of “radius” $r$ fits inside a region iff its centre lies in the region eroded by $r$ (e.g. an inner square or inner disc).
Similar solids: scaling linear size by $k$ scales area by $k^2$ , volume by $k^3$ .
Multinomial counts: if a point falls into disjoint zones with probabilities $p_1,\dots,p_k$ ( $\sum p_i=1$ ), then the counts among $n$ independent points are $\mathrm{Mult}(n;p_1,\dots,p_k)$ .
Order-statistic / conditioning: for radius $R$ with density $f_R$ on $[0,r_0]$ from a uniform disc point, $f_R(r) = 2r/r_0^2$ ; condition on one point’s radius to compute probabilities for another.
Buffon-type: reduce to a uniform angle and/or uniform position, then integrate an indicator over the joint density.

Problems