Bertrand's Paradox Calculator
A single question — “What is the probability that a random chord is longer than the side of an inscribed equilateral triangle?” — can have multiple valid answers. This calculator shows how the sampling method changes the result.
Probability
33.33%
Chord longer than triangle side
Random Endpoints on the Circumference
Choose two points independently and uniformly on the circumference of the circle, then connect them to create a chord.
How the probability arises
The chord is longer than the side of the inscribed equilateral triangle when the arc between the points subtends more than 120°.
| Method | Construction | Probability |
|---|---|---|
| Random Endpoints | Choose two points uniformly on the circumference and connect them. | 33.33% |
| Random Radius Point | Pick a random radius and point along it; draw the perpendicular chord. | 50.00% |
| Random Midpoint | Select a random point inside the circle as chord midpoint. | 25.00% |
How to Use This Calculator
- Select one of the three classic random chord sampling procedures.
- Review the probability output to see how likely a randomly generated chord exceeds the triangle side.
- Compare the construction steps and formulas for each method to understand how the answers diverge.
- Use the summary table to contrast all methods side-by-side.
Formula
The probability depends on how “random chord” is defined. Each method weights the space of chords differently:
- Random Endpoints: P = length of favorable arc ÷ circumference = 1/3.
- Random Radius Point: P = area of inner disk (radius R/2) ÷ area of circle = 1/2.
- Random Midpoint: P = area ratio squared = (1/2)2 = 1/4.
The side of the inscribed equilateral triangle equals R√3. Comparing chord lengths reduces the geometry to whether a point falls inside a particular region.
Full Description
Bertrand's Paradox illustrates that probability questions must specify the underlying random experiment. Different but plausible definitions of a “random chord” weight the sample space differently and thereby produce incompatible answers. Each sampling method is consistent internally; the paradox arises only because the informal question hides the sampling assumptions.
This tool stresses the importance of modelling. When posed without context, the question has no single correct answer. In practice, you should choose the sampling mechanism that matches the application (for instance, how a physical experiment would generate chords) and stick with that interpretation.
Frequently Asked Questions
Is any method “correct”?
Each method is correct once the experiment is specified. The paradox shows that without stating the sampling procedure, the problem is ambiguous.
Why do two methods give the same geometry ratio?
The random radius and random midpoint methods both compare areas of concentric disks. However, the former chooses points along a line segment, while the latter picks midpoints across the full disk. These distinct experiments coincidentally share the same geometric ratio setup but not the same result.
Can I extend this to other shapes?
Yes. The lesson applies broadly: whenever a problem says “select something at random,” you must state the precise sampling distribution. Without that, probabilities remain undefined.
How does this connect to real experiments?
Physical experiments implement one of these methods implicitly. For example, spinning two markers on the circumference mimics the random-endpoints method, while dropping a stick across a circle approximates the random-midpoint model.