A matemática do azar: explorando a ruína do jogador por meio da cadeia de Markov

Abstract

This undergraduate work presents a theoretical and introductory analysis of Markov chains, situating them within the field of stochastic processes. The study begins with a fundamental review of Probability Theory, moving from classical approaches to the consolidation of the axiomatic method proposed by Kolmogorov. Fundamental concepts of random variables and probability distributions are discussed as essentials for understanding the dynamics of systems that evolve over time. The central focus lies on the definition and properties of discrete-time Markov chains, exploring the ’memorylessness’ characteristic and transition matrices. Through classical examples, such as the Gambler’s Ruin, the model’s applicability to random phenomena is demonstrated. Finally, the work briefly discusses future perspectives for expansion into the field of quantum computing via Quantum Random Walks, highlighting the relevance of probabilistic nature in modern science.