Bertrand Paradox Calculator

What is Bertrand's Paradox in Geometric Probability?

Bertrand's Paradox is a fundamental problem in geometric probability theory introduced by Joseph Bertrand in 1889. It demonstrates that probabilities in continuous spaces are not well-defined unless the mechanism that generates random variables is specified.

The Problem Statement

Consider an equilateral triangle inscribed inside a circle. Suppose a chord of the circle is chosen "at random." What is the probability that the chord is longer than a side of the equilateral triangle?

The Three Classical Solutions

1. Method 1: Random Endpoints

Choose two independent points uniformly at random on the circumference of the circle and connect them to form a chord. The probability that the chord exceeds the side length of the inscribed equilateral triangle is:

$$P = \frac{1}{3} \approx 33.33\%$$

2. Method 2: Random Radius

Choose a radius of the circle at random, and then pick a point on that radius uniformly at random to serve as the midpoint of the chord. The probability is:

$$P = \frac{1}{2} = 50.00\%$$

3. Method 3: Random Midpoint

Choose a point uniformly at random anywhere inside the disk of the circle to serve as the midpoint of the chord. The probability is:

$$P = \frac{1}{4} = 25.00\%$$

Why is it a Paradox?

All three methods appear logically valid, yet each yields a different probability. The paradox arises because "at random" is ambiguous when choosing from an infinite continuous set of lines.

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