An Introduction to Probability, Hyper-Probability, and
Super-Hyper-Probability
Takaaki Fujita1,∗, Ajoy Kanti Das2
1Independent Researcher, Tokyo, Japan
2Associate Professor, Department of Mathematics, Tripura University, Agartala-799022, Tripura, India
Emails: Takaaki.fujita060@gmail.com; ajoykantidas@gmail.com
Abstract
Standard probability theory assigns each event a single real value in [0, 1], satisfying non-negativity, nor-
malization, and countable additivity. Hyper-Probability extends this notion by assigning to each event a
set of probability values in [0, 1], thereby capturing multiple independent assessments from diverse sources.
Super-HyperProbability further generalizes the framework by mapping events to iterated power sets of [0, 1],
modeling hierarchical uncertainty across multiple aggregation levels. In this paper, we formally define the
Hyper-Probability Measure and Hyper-Probability Distribution, examine their fundamental properties, and
demonstrate how these constructs unify and extend classical probability within the Hyper- and Super-HyperProbability
paradigms.
Keywords: Probability; HyperProbability; SuperHyperProbability; Probability Distribution; Probability Mea-
sure