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Bertrand Chord Probability Model Calculator — result sheet
Compare the classic probabilities for a chord longer than an inscribed equilateral-triangle side under three explicit random-chord models.
Inputs used
Results
Visual chart
Breakdown
Calculation steps
Returned data table
Formula and methodology
Formula: The threshold chord is √3R; the probability that the chord is longer is 1/3 for two random endpoints, 1/2 for a uniformly selected center distance, and 1/4 for a uniformly selected midpoint over the disk.
The famous chord problem does not have one model-free numerical answer because the phrase choose a chord at random does not specify a probability distribution. This page makes the selection rule the main input and keeps the circle radius visible even though the probability is scale-invariant. The comparison is a lesson about modeling assumptions, not a claim that one of the three mechanisms is universally correct.
This result follows the calculator's declared inputs, precision, validation boundaries, and model limits.
Input contract
- Chord-selection model — 3 choices
- Circle radius — units; minimum 1.0E-6; maximum 1000000
Worked example
| Input | Value |
|---|---|
| Chord-selection model | two-points |
| Circle radius | 1 |
The threshold is √3 ≈ 1.732 units and the two-random-endpoints model gives probability 1/3 ≈ 33.33%.
Assumptions and limits
- The reference figure is a circle with an inscribed equilateral triangle, so the threshold chord length is √3 times the radius.
- Two-points mode selects two circumference points independently with uniform angular position.
- Uniform-distance mode treats the perpendicular distance from the center as uniform from zero to the radius.
- Uniform-midpoint mode treats the chord midpoint as uniformly distributed by area over the disk.
- The chord is longer when its center distance is less than half the radius.
- The page compares ideal mathematical selection rules and does not model a physical drawing device or measurement error.
Calculator note
Source and methodology
Use the official WorldCalculate methodology policy for the source, formula, precision, and boundary standards behind this calculator.
Planning estimate, not financial, medical, legal, or professional advice. © WorldCalculate — reuse with attribution. Built and curated by Hassan ALRowaie.
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