Nonelementary Notes on Elementary Events

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.