Bertrand's Paradox
Bertrand's Paradox calculator. Random chords in a circle: Method 1 gives 1/3, Method 2 gives 1/2, Me
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
Bertrand's Paradox โ Random Chord: 1/3, 1/2, or 1/4
Same question, three valid answers. Method 1 (endpoints): 1/3. Method 2 (radial): 1/2. Method 3 (midpoint): 1/4. "Random" is ambiguous without a selection method.
Examples โ Click to Load
Selection Method
Visual: Circle, Triangle & Sample Chords
Green = long chord, Red = short chord
Method 1: Two random points on the circle
Theoretical P(long chord) = 0.3333333333333333 (33%)
Simulation
Method Comparison โ All Three Probabilities
Simulation Results โ Long vs Short Chords
โ ๏ธFor educational and informational purposes only. Verify with a qualified professional.
๐ Key Takeaways
- โข The same question "what is a random chord?" gives 3 different valid answers depending on the selection method
- โข This paradox shows that probability requires a well-defined sample space โ "random" alone is ambiguous
- โข Method 1 (endpoints): P=1/3, Method 2 (radial): P=1/2, Method 3 (midpoint): P=1/4
- โข The paradox is resolved by noting each method defines a different probability measure on the space of chords
- โข Modern resolution: Jaynes (1973) argued Method 2 (P=1/2) is the "natural" answer based on maximum entropy
๐ก Did You Know
๐ How It Works
1. The Setup
Equilateral triangle inscribed in a circle. A chord is longer than a triangle side if it subtends an arc > 120ยฐ. We ask: P(random chord > side).
2. Method 1: Random Endpoints โ P=1/3
Fix one point on the circle, choose another uniformly. Chord is long if the other point falls in the "long chord" arc (1/3 of circumference).
3. Method 2: Random Radial Point โ P=1/2
Choose a random radius, pick a random point on it, draw perpendicular chord. Chord is long if the point is closer to center than the midpoint of a triangle side (which bisects the radius).
4. Method 3: Random Midpoint โ P=1/4
Choose a random point inside the circle as the chord's midpoint. Chord is long if midpoint falls within inner circle of radius r/2. Area ratio = ฯ(r/2)ยฒ/(ฯrยฒ) = 1/4.
๐ฏ Expert Tips
Always define your sample space
Probability without a well-defined measure is meaningless
Jaynes' resolution
Maximum entropy suggests Method 2 when you have no prior information
Physical verification
Drop spaghetti on a circle โ it empirically supports P=1/2
This generalizes
The lesson applies to any continuous probability: specify HOW randomness is generated
โ๏ธ Method Comparison
| Aspect | Method 1 | Method 2 | Method 3 |
|---|---|---|---|
| Construction | Random endpoints on circle | Random point on random radius | Random midpoint in disk |
| Probability | 1/3 | 1/2 | 1/4 |
| Physical meaning | Pick 2 points on rim | Drop perpendicular from radius | Throw dart at disk |
โ Frequently Asked Questions
Which answer (1/3, 1/2, or 1/4) is "correct"?
All three are correct under their respective selection procedures. The paradox shows that 'random chord' is underspecified โ you must define HOW the chord is chosen.
What does this paradox teach us about probability?
Probability requires a well-defined sample space and measure. 'Random' alone is ambiguous for continuous spaces.
How did Jaynes resolve the paradox?
Jaynes (1973) used the maximum entropy principle: with no prior information, Method 2 (P=1/2) maximizes entropy and is the 'natural' choice. Physical experiments support this.
What is geometric probability?
Geometric probability assigns probabilities based on geometric measures (length, area, volume). Bertrand's paradox is a classic example where different geometric parameterizations give different answers.
How do physical experiments relate to the paradox?
Dropping straws or needles onto a circle naturally implements something like Method 2. Experiments consistently yield P โ 1/2, supporting Jaynes' resolution.
Is this related to the Monty Hall problem?
Both involve careful specification of randomness. Monty Hall is about conditional probability; Bertrand's paradox is about defining the sample space. Different lessons, but both show that 'random' needs precision.
What is the maximum entropy principle?
When you have no prior information, choose the distribution that maximizes entropy (uncertainty). Jaynes argued this singles out Method 2 as the 'least informative' prior.
How does this paradox apply to real-world problems?
Whenever you model 'random' selection in continuous spaces (e.g., random sampling, Monte Carlo methods), you must specify the distribution. The paradox is a cautionary tale for statisticians and modelers.
๐ Infographic Stats
๐ Official Sources
โ ๏ธ Disclaimer: This calculator demonstrates Bertrand's chord paradox for educational purposes. Each method is mathematically valid; the paradox illustrates that "random" must be precisely defined. Simulations are subject to random variation.