Stochastic Models for Breaking Large Bills and Coins

Annual Meeting in Mathematics 2024

Abstract

This paper introduces stochastic models, specifically employing Markov chains, to analyze the dynamics of breaking down large bills and coins. We derived transition matrices and faced challenges in manually solving the equilibrium distribution. The investigation highlights the properties of all transition matrices, characterized by irreducibility, positive recurrence, and aperiodicity, ensuring the existence of limiting probabilities. However, the complexity of solving the system of equations for these probabilities prompted the utilization of a simulation study for approximation. Results indicate that probabilities associated with each state, determined by the last digit of the held money, are approximately equal. Furthermore, when multiple ways exist to carry money for a given last digit, the study provides insights into the proportional occurrence of each subcase.