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4 Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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Fuzzified probability : from Kolmogorov to Zadeh and beyond

Frič R., Papčo M.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2014

Vol. 19

179--188

We discuss the fuzzification of classical probability theory. In particular, we point out similarities and differences between the so-called fuzzy probability theory and the so-called operational probability theory.

Random walk - fuzzy aspects

Frič R., Papčo M.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2011

Vol. 16

205--212

Some beautiful and powerful mathematical ideas are hard to present to students because of the involved abstract language (notation, definitions, theorems, proofs, formulas) and lack of time. Animation and “mathematical experiments” provide a remedy. In the field of stochastics, the Galton board experiment presents several fundamental stochastic notions: a random event, independent random events, the binomial distribution, limit distribution, normal distribution, interpretation of probability, and leads to their better understanding. Random walk is a natural generalization of the Galton board. We use random walks as a motivation and presentation of basic principles of fuzzy random events and fuzzy probability. Fuzzy mathematics and fuzzy logic generalize classical (Boolean) mathematics and logic, reflect everyday experience and decision making and have broader applications. Experimenting with random walks also sheds light on the transition from classical to fuzzy probability.

Triangular structures and duality

Frič R., Papčo P.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2007

Vol. 12

23-28

We introduce and study the category AFD the objects of which are generalized convergence D-posets (with more than just one greatest element) of maps into a triangle object T and the morphisms of which are sequentially continuous D-homomorphisms. The category AFD can serve as a base category for antagonistic fuzzy probability theory. AFD-measurable maps can be considered as generalized random variables and ADF-morphisms, as their dual maps, can be considered as generalized observables.

Nonelementary Notes on Elementary Events

Frič R.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2003

Vol. 9

137--141

Our goal is to present simple examples illustrating the nature and role of elementary events and random variables in probability theory, both classical and operational (fuzzy). As stated in Płocki [10], in teaching probability we should concentrate on the construction of probability spaces and their properties, and not on the calculation of probability of various strange events (like hitting a bear if we can shoot three times, etc.). On a rather advanced level, Łoś [8] analyzed the constructions of probability spaces in the classical probability. J. Loś explained the nature and underscored the role of elementary events. Roughly, the events form a Boolean algebra, but some probability properties of the algebra depend on its representation via subsets and this is done via the choice of some fundamental subset of events and the choice of elementary events. Remember, choice! There are situations in which the classical probability model is not quite suitable (quantum physics, fuzzy models, c.f. Dvurečenskij and Pulmannová [3], Frič [5]), and I would like to present simple examples and simple models of such situations. In order to understand the generalizations, let me start with a well-known example of throwing two dice.