Bertrand's Paradox Calculator
Explore the famous probability paradox with three different interpretations
Bertrand's Paradox Simulation
Different ways to generate random chords
Radius of the circle (default: 1)
More simulations provide more accurate results (100 - 100,000)
Results
Method: Random Endpoints
Triangle side length: 1.7321 units
Circle radius: 1 units
Current Method: Random Endpoints
One endpoint is fixed on the circle, and the other is randomly placed on the circumference.
Chords are longer than the triangle side when the second endpoint falls in the arc opposite to the first endpoint (1/3 of the circumference).
Expected probability: 33.33%
The Three Solutions to Bertrand's Paradox
1. Random Endpoints Method
Fix one endpoint and randomly select the other on the circle's circumference.
Probability: 33.33% (1/3)
2. Random Radial Point Method
Select a random point on a radius and draw a chord perpendicular to it.
Probability: 50% (1/2)
3. Random Midpoint Method
Select a random point inside the circle as the chord's midpoint.
Probability: 25% (1/4)
Understanding the Paradox
Setup
Circle with inscribed equilateral triangle
Question
What's the probability a random chord is longer than the triangle's side?
Paradox
Three valid methods give three different answers
Key Concepts
Equilateral triangle side = r√3
Principle of indifference leads to ambiguity
Different sampling methods = different results
All three answers are mathematically correct
Understanding Bertrand's Paradox
What is Bertrand's Paradox?
Bertrand's paradox is a problem in geometric probability, first posed by Joseph Bertrand in 1889. It demonstrates that the concept of "random" can be ambiguous when dealing with infinite sets.
The Problem Statement
Given a circle with an inscribed equilateral triangle, what is the probability that a randomly chosen chord of the circle is longer than a side of the triangle?
Why Three Different Answers?
The paradox arises because there are infinitely many chords in a circle, and the word "random" doesn't specify how to choose among them. Different reasonable interpretations of "random chord" lead to different probability distributions and thus different answers.
Mathematical Foundation
Circle radius: r
Triangle side length: r√3
Circle area: πr²
The Three Methods
Random Endpoints
P = 1/3 ≈ 33.33%
Random Radial Point
P = 1/2 = 50%
Random Midpoint
P = 1/4 = 25%
The Resolution
Bertrand's paradox illustrates that when dealing with infinite sets, additional information is needed to define what "random" means. Each of the three methods represents a different way of uniformly distributing chords, and each is mathematically valid.
The paradox demonstrates the importance of clearly defining the sample space and probability measure in geometric probability problems. In practice, the "correct" answer depends on the specific physical or mathematical context of the problem.