Bertrand's Paradox

The paradox arises when we try to find the probability that a randomly chosen chord of a given circle is longer than the side of an inscribed equilateral triangle. Depending on how the randomness is generated, the probability evaluates to 1/3, 1/2, or 1/4. The three primary methods are the Random Endpoint method (yielding 1/3), the Random Radius method (yielding 1/2), and the Random Midpoint method (yielding 1/4). The underlying math simulates these geometric selections across a large sample of numbers to contrast the theoretical probabilities against simulated outputs.

Imagine running a simulation with a sample size (number sim) of 10,000 trials. For the Random Endpoint method, suppose the simulation counts 3,350 successful chords where the length exceeds the inscribed triangle side. The probability is calculated as prob0 = goodangle / number_sim, resulting in 3,350 / 10,000 = 0.335. Comparing this to the theoretical value of 0.3333 gives a tiny margin of error, illustrating how convergence works in geometric probability.

When studying geometric probability, always define your sample space explicitly before running simulations. Recognize that "randomness" is not a universal constant; it requires a precise measure or coordinate system. Adjusting your sample size higher will reduce statistical variance and align your experimental probabilities closer to their exact theoretical limits.