Removing the inherent paradox of the Buffon´s needle Monte Carlo simulation using Fixed-Point iteration method

In teaching simulation, the Buffon´s needle is a popular experiment to use for designing a Monte Carlo simulation to approximate the number π. Simulating the Buffon´s needle experiment is a perfect example for demonstrating the beauty of a Monte Carlo simulation in a classroom. However, there is a common misconception concerning the Buffon´s needle simulation. Erroneously, the simulation of the needle drop cannot be used to evaluate π. We have to simulate the needle´s angle from an uniform (0, π over 2) distribution. It is self-referential in theory, since it requires the number π as the input value to approximate π. In this study, we propose a new method using the fixed-point iteration to remove the inherent paradox of the Buffon´s needle simulation. A new algorithm with Python implementation is proposed. The simulation outputs indicate that our new method is as good as if we use the true π value as an input.