Volume 6 • Issue 1 • PP: 01-21 • 2026
An Introduction to Probability, Hyper-Probability, and Super-Hyper-Probability
View PDF Abstract
Standard probability theory assigns each event a single real value in [0, 1], satisfying non-negativity, normalization, 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
References
[1] Melody Mae Cabigting Lunar and Renson Aguilar Robles. Characterization and structure of a power set graph. International Journal of Advanced Research and Publications, 3(6):1–4, 2019.
[2] Florentin Smarandache. Foundation of superhyperstructure & neutrosophic superhyperstructure. Neutrosophic Sets and Systems, 63(1):21, 2024.
[3] MA Shalu and S Devi Yamini. Counting maximal independent sets in power set graphs. Indian Institute of Information Technology Design & Manufacturing (IIITD&M) Kancheepuram, India, 2014.
[4] Jacob Stegenga. The natural probability theory of stereotypes. Diametros, 22(83):26–52, 2025.
[5] Himadri Deshpande. Foundations of Probability Theory. Educohack Press, 2025.
[6] Takaaki Fujita. An introduction and reexamination of hyperprobability and superhyperprobability: Comprehensive overview. Asian Journal of Probability and Statistics, 27(5):82–109, 2025.
[7] Freddy Delbaen. Coherent risk measures on general probability spaces. 2002.
[8] Jacob Burbea and C. Radhakrishna Rao. Entropy differential metric, distance and divergence measures in probability spaces: A unified approach. Journal of Multivariate Analysis, 12:575–596, 1982.
[9] Robert B Ash and Catherine A Dol´eans-Dade. Probability and measure theory. Academic press, 2000.
[10] John S Ramberg, Edward J Dudewicz, Pandu R Tadikamalla, and Edward F Mykytka. A probability distribution and its uses in fitting data. Technometrics, 21(2):201–214, 1979.
[11] AJC Wilson. The probability distribution of x-ray intensities. Acta Crystallographica, 2(5):318–321, 1949.
[12] David A Nix and Andreas S Weigend. Estimating the mean and variance of the target probability distribution. In Proceedings of 1994 ieee international conference on neural networks (ICNN’94), volume 1, pages 55–60. IEEE, 1994.
[13] Kianoush Nazarpour, Ali H Al-Timemy, Guido Bugmann, and Andrew Jackson. A note on the probability distribution function of the surface electromyogram signal. Brain research bulletin, 90:88–91, 2013.
[14] John Gordon Skellam. A probability distribution derived from the binomial distribution by regarding the probability of success as variable between the sets of trials. Journal of the Royal Statistical Society. Series B (Methodological), 10(2):257–261, 1948.
[15] M Edimu, CT Gaunt, and R Herman. Using probability distribution functions in reliability analyses. Electric Power Systems Research, 81(4):915–921, 2011.
[16] Jismi Mathew and Milin K Anil. A retrospective review of biomedical applications of neutrosophic probability distributions. Asian Journal of Probability and Statistics, 28(2):66–79, 2026.
[17] Hina Khan, Kanwal Javid, et al. A proposed neutrosophic probability model for normalized differencevegetation index using remote sensing: Model building on climate. In Multi-Criteria Decision Making Models and Techniques: Neutrosophic Approaches, pages 205–226. IGI Global Scientific Publishing, 2025.
[18] Suman Das, Bimal Shil, Rakhal Das, Huda E Khalid, and AA Salama. Pentapartitioned neutrosophic probability distributions. Neutrosophic Sets and Systems, 49:32–47, 2022.
Cite This Article
Choose your preferred format