Monte Carlo method in simulations

Introduction

Monte Carlo simulation is a mathematical technique used to simulate the possible outcomes of an uncertain event. Instead of fixed inputs, a probability distribution is used as a parameter at at least one point. By generating random probabilities as an input, a set of different outcomes and the probability of this outcomes can be generated through repeated simulation. The generation of random numbers by computers plays an important role in this.

The first well-known experiment that used randomness to solve a problem was "Buffon's Needle". The French scientist Georges Louis LeClerc, Comte de Buffon asked in the 18th century the question with what probability a randomly thrown needle intersects a grid of parallel lines. Only a few years later he was able to prove that this probability is 2L/Dπ (D > L). Let L be the length of the needle and D the distance between the lines. In his first experiments, LeClerc allegedly threw baguettes over his shoulder onto a tiled floor. In 1812, the French scientist Pierre-Simon Laplace suggested that this experiment could be used to determine the number π. In 1901, the scientist Mario Lazzarini was able to determine the number π to 6 decimal places by throwing needles on a grid of parallel lines, and that after only 3408 throws. However, this result is also doubted by scientists or also described as a stroke of luck. This is because in order to determine the number π to 6 decimal places in a 95% confidence interval, one would theoretically have to throw at least 134 trillion needles. Nowadays, these experiments can be simulated much faster and more efficiently with the help of computers. This is explained in detail in the chapter "Monte Carlo Simulation and the Number Pi".

The previous example of "Buffon's Needle" is based on actual chance by throwing a needle. Moreover, this experiment explored a deterministic problem that had already been solved analytically. Today's Monte Carlo simulations, however, approach the process in reverse. They serve as a tool when complex problems cannot be solved analytically. In addition, the simulations use artificially or computer-generated randomness. This was first used by John von Neumann and Stanislaw Ulam to explore the properties of neutron travel through radiation shielding. In doing so, they contributed to the invention of the atomic bomb during the Manhattan Project. They also gave the name to the Monte Carlo simulation, named after the famous casino in Monaco. Most of the Monte Carlo principles used today were developed during research into nuclear weapons.