This paper presents a novel framework for probability theory, leveraging the geometric partitioning of Voronoi diagrams and topological measure theory to derive the fundamental properties of probability—nonnegativity, normalization, and additivity—without relying on the traditional Kolmogorov axioms. We reveal the abstract nature of probability: it is independent of specific entities and determined solely by the squared modulus of complex coefficients |ci|2. By treating Voronoi cells as phase circles and mapping them to Hilbert space basis vectors, we provide a geometric interpretation for classical probability and a topological foundation for quantum probability, expressed as P(|ei⟩) = |ci|2. We extend this framework to continuous distributions using three-dimensional tubular structures, and incorporate conditional probability and Bayesian inference, demonstrating its versatility. This framework unifies classical and quantum probability, highlighting the universal nature of probability, with potential applications in quantum information processing, random geometry, and statistical physics.