A circular wooden frame with loosely arranged threads in a conceptual probability studio. Original AI-generated artwork produced for GreenCalc; not a measured diagram or empirical data display.
“Random chord” leaves a missing instruction
A chord is longer than the side of an inscribed equilateral triangle when its length exceeds √3 times the circle radius. Bertrand's paradox asks for the chance of that event. Different random-chord mechanisms give different answers because they distribute probability across chords differently. There is no single answer until the sampling rule is stated.
Compare three concrete mechanisms
With two independent uniformly selected boundary endpoints, the probability is one-third. With a uniformly selected perpendicular distance from the center, it is one-half. With a midpoint selected uniformly over disk area, it is one-quarter. In all three mechanisms the circle and threshold are identical; the distribution of chord distance from the center changes.
| Sampling rule | Probability longer than √3 R |
|---|---|
| Uniform boundary endpoints | 1/3 |
| Uniform perpendicular distance | 1/2 |
| Uniform midpoint over disk area | 1/4 |
Use geometry to check the event
A chord whose perpendicular distance from the center is d has length 2√(R²−d²). Requiring a length greater than √3 R is equivalent to d < R/2. Uniform distance therefore puts half its probability below R/2. A midpoint uniform over disk area falls inside radius R/2 with probability (R/2)²/R² = 1/4. The endpoint mechanism has its own angle distribution and produces the one-third result.
Match the calculation to a reproducible experiment
The Bertrand's Paradox calculator lets you choose the mechanism and vary the threshold-to-radius ratio. It is a probability comparison, not a claim that one mechanism is universally the correct definition of randomness. If implementing a simulation, describe exactly how endpoints, distance or midpoint are sampled. The uniform-radius guide explains a common source of accidental mismatch. For the unrelated coin-box conditioning puzzle, use Bertrand's Box Paradox.
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