Conditional and Non-Standard Probability

In the classical theory of probability we start by a pair (Ω, ), with Ω set of elementary events and σ-algebra of subsets of Ω, called events. A probability on is defined as a countable additive and positive function p such that p(Ω)=1. If A and B are two events, the conditional probability p(A/B) is defined if and only if p(B) is not null, by considering the quotient p(AB)/p(B). In the theory of finitely additive probability, that generalizes the classical one and that is the fundament of the subjective probability, the conditional probability p(A/B) has a meaning also if B has null probability, with the only condition that B is different from the impossible event. But, in this case, we don’t have a relation between p(A/B), p(AB) and p(B). In this paper we prove that such relation and many others can be find if we use the new concepts of probability and conditional probability with values on a hyperstructure. KEYWORD: Finitely additive probabilities. Conditional probabilities. Hypergroups. Hyperrings. Probability with values on a hyperstructure. Non standard measures. Qualitative measures.