Bertrand's Paradox (1889) asks: for a random chord of a circle, what is the probability it is longer than the side of the inscribed equilateral triangle? The "obvious" question turns out to have three different well-defined answers, because "random chord" can mean three different (equally reasonable) sampling procedures:
Chord favorable ⇔ length > R√3 ⇔ distance from center to chord < R/2
Method 1 (Endpoints): both chord endpoints drawn
independently, uniformly on the circle. P = 1/3
Method 2 (Radial Point): pick a random radius
direction, then a point uniform along [0,R] on
it; the chord passes through that point,
perpendicular to the radius. P = 1/2
Method 3 (Midpoint): pick a point uniformly by
AREA inside the disk; the chord has that point
as its midpoint (perpendicular to the radius
through it). P = 1/4
All three procedures produce "a uniformly random chord" by a reasonable-sounding definition, yet they generate different probability distributions over chord length — because "uniform" depends on which parameter (endpoint angle, radial offset, or 2D area) you make uniform. Bertrand used this to argue that "randomness" is not well-defined until the sampling procedure is fixed. Run enough trials on each method here and the running estimate visibly converges to 1/3, 1/2, and 1/4 respectively, exactly as the Law of Large Numbers predicts for each method's own distribution.