A Note on Fuzzy Probability of a Fuzzy Event

Abstract

This article modifies evaluating the probability of a fuzzy event based on a classical probability space introduced by Yager (Inform. Sci. 1979;18:113–122). The presented theoretical results will be illustrated with some propositions and remarks. It is shown that a modification in the definition of fuzzy probability of fuzzy events into a concept of such probabilities has some desirable properties from mathematical points of view. Then, the proposed fuzzy probability will be compared to that of the similar proposed methods. The proposed fuzzy probability will be illustrated with some numerical examples.

Intuitionistic Fuzzy Probability of an Event

“…Here, inspiring by Hesamian and Shams [18], we extend a notion of converging a sequence of IFSs. We will use it to investigate the properties of the proposed intutionistic probability of an event in the next section.…”

“…Suppose that the cation exchangeable capacity, X, is characterized by a normal distribution with with mean µ = (17; 1.50, 1.50; 3.50, 3.50) T and variance σ 2 = 9. We wish to calculate the intuitionistic probability of A = [13,18]. To do so, from equations (3.1) and (3.2), we get:…”

“…In such cases, fuzzy set theory successfully provides suitable tools for modeling probability of a uncertain event. The topic of probability theory with fuzzy information has been studied by many authors during the last decades [3,4,11,8,9,16,17,18,19,20,21,23,24,25,27,28,30,31,33,34,35,36,37,38,39,40,41,43,44]. The other resent approaches to fuzzy probability have been recently proposed are based on the concepts of sets of probability measures, distribution envelops, interval probabilities, p-box approach, and Dempster-Shafer theory (for instance, refer to [5,15].…”

Modeling random and non-random decision uncertainty in ratings data: a fuzzy beta model

“…When a probability space is defined over the reals, the probability of a fuzzy set ℙ(Ã) can also be defined. Over the years, there have been various attempts to define the probability of a fuzzy set in terms of expected value of its membership function (Zadeh 1968), conditional probability of prior information (Coletti and Scozzafava 2004;Singpurwalla and Booker 2004), imprecise probability (Augustin et al 2014), fuzzy numbers (Hesamian and Shams 2017), and likelihood induced by random events (Cattaneo 2017). Following the findings of Denoeux (2011), in this contribution we adopt Zadeh's definition of fuzzy probability (Zadeh 1968).…”

Modeling random and non-random decision uncertainty in ratings data: A fuzzy beta model

“…When a probability space is defined over the reals, the probability of a fuzzy set P( Ã) can also be defined. Over the years, there have been various attempts to define the probability of a fuzzy set in terms of expected value of its membership function [83], conditional probability of prior information [18,71], imprecise probability [26], fuzzy numbers [40], and likelihood induced by random events [14]. Following the findings of [24], in this contribution we adopt Zadeh's definition of fuzzy probability [83].…”