Abstract
This chapter focuses on three principal concepts of probability theory that have been expressed throughout the history of the subject. First, the classical theory is founded on the indefinable concept of equally likely events. Second, the limit-of-relative-frequency theory is founded on an observational concept and a mathematical postulate. Third, the logical theory defines probability as the degree of confirmation of a hypothesis with respect to an evidence statement. The classical theory considers the mutually exclusive, exhaustive cases, with the probability of an event defined as the ratio of the number of favorable cases to the total number of possible cases. The logical theory is capable of dealing with certain interesting hypotheses, yet its flexibility is academic and generally irrelevant to the solution of gambling problems. Gambling phenomena frequently require the direct extension of probability theory axioms and corollaries into the realm of permutational and combinatorial analysis. A permutation of a number of elements is any arrangement of these elements in a definite order and a combination is a selection of a number of elements from a population considered without regard to their order.
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