Bernoulli Trials: From a Fuzzy Measure Point of View

Bernoulli ("Ars Conjectandi," Basle, 1713) proved the first limit theorem of law of large numbers which provided the foundation of probability and statistical theory. However, the problem of Bernoulli trials is still unsettled (e.g., see Hacking, "The Emergence of Probability," Cambridge Univ. Press, Cambridge, 1975). It is from different interpretations of the relationship between the Bernoulli trials and relative frequency that we have different schools of probability theories (e.g., see Cox (Amer. J. Phys.14, No. 1 (1946), 1-13) and Fine (IEEE Trans. Inform. TheoryIT-16, No. 3 (1970), 251-257)). In this paper we give a new treatment of the Bernoulli trials based on fuzzy measure, and we interpret the Bernoulli trials through the interaction of probability and possibility measures.