Probability interpretations
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The word probability has been used in a variety of ways since it was first coined in relation to games of chance.
There are two broad categories of probability interpretations: Frequentists assign probabilities only to random events according to their relative frequencies of occurrence or to subsets of populations as proportions of the whole. Bayesians assign probabilities to any statement, even when no random process is involved, as a way to represent its plausibility. As such, the scope of Bayesian inquiries include the scope of frequentist inquiries.
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Epistemological controversy
While frequentism is widely accepted as a scientific tool, the use of Bayesian probability often raises the philosophical debate as to whether it can contribute valid justifications of belief.
In order to resolve a particular problem using probability theory, Bayesians will accept to apply a particular probability model, such as the urn model, to a thought experiment. The issue is that for a given problem, multiple thought experiments could apply, and choosing one is a matter of intuition or belief. The "sunrise problem"" illustrates the issue. A particular version of this issue is the reference class problem.
Practical controversy
The difference of view has also many implications for the methods by which statistics is practiced, and for the way in which conclusions are expressed. When comparing two hypotheses and using some information, frequency methods would typically result in the rejection or non-rejection of the original hypothesis with a particular degree of confidence, while Bayesian methods would suggest that one hypothesis was more probable than the other.
As a possible solution, the eclectic view accepts both interpretations: depending on the situation, one selects one of the 2 interpretations for pragmatic, or principled, reasons.
Axiomatic probability
The mathematics of probability can be developed on an entirely axiomatic basis that is independent of any interpretation: see the articles on probability theory and probability axioms for a detailed treatment.