What is the probability that a random chord of a circle is longer than the side of an equilateral triangle inscribed in the circle?

What is the probability that a random chord of a circle is longer than the side of an equilateral triangle inscribed in the circle?

Approach: Depends on interpretation of 'random chord'. Different reasonable interpretations yield different answers.

1. Random endpoints: chord length = 2R sin(θ/2), side length = R√3. Probability = 1/3.,2. Random midpoint: probability = 1/4.,3. Random radius and point: probability = 1/2.

Follow-up: Explain why this is called Bertrand's paradox.

Key concepts: Bertrand paradox, measure theory, geometric probability.