Entropy metrics of biased multisided dice probability distributions

Abstract

Entropy is one of the most fundamental concepts in physics and information theory (Claude E. Shannon (1948), \cite{Shannon}), its correct understanding is essential for the study of various physical systems, especially in thermodynamics, statistical mechanics, and in thermal process engineering. This blending of concepts from physics and information theory has made it difficult for many students, even specialists to clearly grasp the power, scope, and proper domain of applicability of the theory, particularly at the undergraduate and graduate levels, mainly due to the abundance of highly artificial theoretical models. In this work, we present a systematic and comprehensive study of the entropy applicability, starting from elementary random systems such as fair (unbiased) and biased dice throws. Specifically, we examine single dice with an arbitrary number of faces, i.e., N=s. We build upon previous studies (\cite{SanchezQ1,SanchezQ2}) that serve as a basis to develop a mathematically consistent analysis, while avoiding ambiguities and unnecessary formal excesses. We compute the corresponding entropy measures for generalized cases involving biased distributions using the probability functions derived in these previous works —additionally the Escher-type distribution for demonstrating its equivalence with the Gibbs–Boltzmann distribution— from which the unbiased cases naturally arise as particular limits. Thus, we provide more significant insight into both the analytical development and the visual interpretation of the resulting entropy metrics.\\ This theoretical framework can be used to introduce advanced undergraduate or postgraduate students to the fundamental characteristics of biased random systems (distributions) and their intrinsic entropy, which are more general than standard cases.