On the dynamical evolution of randomness Part A: Random experiments as dynamical systems

TL;DR

This paper extends Bernoulli's Law of Large Numbers to explore how empirical randomness evolves in repeated experiments like coin tosses and dice rolls, revealing dynamical behaviors and the influence of chance on randomness.

Contribution

It introduces a dynamical systems perspective to empirical randomness, extending classical probability theorems with analytical and numerical insights into the evolution of randomness.

Findings

Empirical randomness increases with chance depending on probability growth rate

Dynamical behaviors of random experiments are characterized and verified

Analytical and simulation results support the extended theorem

Abstract

In this work, Bernoulli's Law of Large Numbers, also known as the Golden theorem, has been extended to study the relations between empirical probability and empirical randomness of an otherwise random experiment. Using the example of a coin toss and a dice role, some interesting results are drawn. Analytically and using numerical computations, empirical randomness of each outcome has been shown to increase by \textit{ chance}, which itself depends on the growth rate of empirical probabilities. The analyses presented in this work, apart form depicting the nature of flow of random experiments in repetitions, also present dynamical behaviours of the random experiment, and experimental and simulation-based verifications of the mathematical analyses. It also presents an appreciation of the beauty of Bernoulli's Golden theorem and its applications by extension.