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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 talk about probabilities only when dealing with well defined random experiments. The relative frequency of occurrence of an experiment's outcome, when repeating the experiment, is a measure of the probability of that random event. Bayesians, on the other hand, assign probabilities to any statement whatsoever, even when no random process is involved, as a way to represent its subjective plausibility.
Bayesians point to the work of Ramsey and de Finetti as proving that subjective beliefs must follow the laws of probability if they are to be coherent.
The use of Bayesian probability involves specifying a prior probability. This may be obtained from consideration of whether the required prior probability is greater or lesser than a reference probability associated with an urn model, or a thought experiment. The issue is that for a given problem, multiple thought experiments could apply, and choosing one is a matter of judgement. Thus different people may assign different prior probabilities. The "sunrise problem" illustrates the issue. A particular version of this issue is the reference class problem.
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.
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.
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