Bertrand's Chord Paradox and Invariant Measures

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Continuous RVs / Bertrand's Paradox.

Your Task
A chord is chosen at random in a circle of radius $R$. An equilateral triangle is inscribed in the circle. The side length of the triangle is $\sqrt{3}R$. Calculate the probability that the chord is longer than $\sqrt{3}R$ under the three classical interpretations:

1. Random endpoints: Two points chosen uniformly and independently on the circumference.
2. Random radius: A radius is chosen uniformly at random, and a distance $r \in [0, R]$ along that radius is chosen uniformly.
3. Random midpoint: A point is chosen uniformly at random from the interior of the disc.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.