Monte Carlo simulations as a route to compute probabilities

TL;DR

This paper demonstrates how Monte Carlo simulations, implemented in MATLAB, can be used to model probability problems, providing insights into systems with and without analytical solutions, especially for educational purposes.

Contribution

It introduces Monte Carlo simulation techniques for probability problems at high school and undergraduate levels, emphasizing educational applications and system modeling.

Findings

Simulations help visualize probability concepts.

Applicable to systems lacking analytical solutions.

MATLAB's rand function effectively generates randomness.

Abstract

Monte Carlo simulations are based on the manipulation of random numbers to evaluate probable outcomes, with applicability in a variety of different fields. By assigning probabilities, which can be determined a priori, to various events, it is possible to track the evolution of the system over length and time scales which are not normally accessible to other simulation techniques. Monte Carlo simulations can provide insights, which can be used to develop more realistic models. In this work, these simulations are used to model a variety of probability problems normally encountered at the high school and undergraduate level. The simulations are used to introduce concepts related to system size, simulation runs (repeatability), and basic statistics. While many of the problems discussed here have analytical expressions, systems where easy analytical solutions are not available are also discussed. In this work, MATLAB is used to model these systems and the rand function in the software is used extensively for random number generation.