Concentric translucent circular plates and a small set of scattered beads. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
Equal radial steps cover unequal areas
Choosing an angle uniformly and a radius uniformly does not place points uniformly over a disk. A ring near the outside contains more area than a ring of the same width near the center. Uniform radius assigns equal probability to those equal-width rings, concentrating probability per unit area toward the center. A scatter plot may look random while still implementing the wrong spatial model.
Derive the correct disk-area radius
For a point uniform over a disk of radius R, the probability of lying within distance r of the center is r²/R². If U is uniform from zero to one, use r = R√U and an independent uniform angle. By contrast, r = RU makes the radius uniform. At R/2, the disk-area model assigns probability one-quarter; the uniform-radius model assigns one-half.
See why this changes chord probabilities
A chord with midpoint distance r has length 2√(R²−r²). For the equilateral-triangle threshold, long chords have midpoint distance below R/2. Uniform disk midpoints therefore give a one-quarter probability, while uniform radial distance gives one-half. The two mechanisms answer different sampling questions even though each draw uses uniform random numbers internally.
Test the transformed variable, not just the picture
When checking a disk sampler, the squared normalized radius r²/R² should behave uniformly under the area model. Under a uniform-radius model, r/R is the uniform quantity. Record the transformation with the code so a later change does not silently alter the intended experiment. Use the Bertrand's Paradox calculator to compare the implied chord probabilities for different thresholds. The companion three-mechanism guide includes the boundary-endpoint model and a geometric threshold check. This spatial sampling issue differs from the birthday collision problem, where the model instead specifies uniform independent draws into discrete buckets.
Check the model behind your result
Use the Bertrand's Paradox for the supported calculation and review its stated assumptions before interpreting the output. Find related tools in the Statistics category, or browse the GreenCalc homepage for other calculation topics.
By GreenCalc Editorial Team · Updated 2026-10-26