Theorem 2.1 holds that is a real parameter which one can validate in the world; conversely theorem 2.2 entails that translates a personal belief into a number. The indefinite series mirrors personal credence upon the repeated event A_{1 }and cannot be compared to which is an authentic measure controllable in the physical reality. Hence it is worth the ponderous verification of since this is a physical measure; instead is a symbol.

It is true that is flexible to countless single situations, but the theorem of a single number proves that does not have any counterpart in the physical reality.

4. The Nature of the Probability

Least we can answer from Theorems 2.1 and 2.2 whether the probability is a measure with physical meaning, or is not physical or even has a double nature.

Theorem 2.1 demonstrates that have different properties respect to, namely the frequentist nature and the subjective nature of probability relay on two separated situations. Therefore (13) and (14) prove to be compatible in point of logic since and refer to distinct physical events. It may be said that is a real, controlled measure when:

(15)

Instead has a personal value when:

(16)

It is reasonable to conclude that the probability is double in nature and professional practice sustains this conclusion as the classical statistics and the Bayesian statistics investigate different kinds of situations. They propose methods that refer to separated environments. To exemplify physicists who validate a general law adopt the Fisher statistics; instead when a physicist has to make decision on a single experiment usually he adopts the Bayesian methods.

5. Conclusions

The interpretation of the probability is considered a tricky issue since decades, and normally theorists tackle this broad argument from the philosophical stance. The philosophical approach yielded several intriguing contributions but did not provide the definitive solution so far. We fear philosophy is unable to bring forth the overwhelming proof upon the nature of the probability because of its all-embracing method of reasoning. Instead the present paper suggests the analytical approach which keeps apart the concept of probability from its argument and examines the influence of A_{1} and A_{n} over P.

Nowadays some statisticians are firm adherents of one or other of the statistical schools, and endorse their adhesion to a school when they begin a project; in a second stage each expert uses the statistical method which relies on his opinion. However this faithful support for a school of thought may be considered an arbitrary act as relying on personal will.

Another accepted view is that each probabilistic theory has strengths and weaknesses and that one or the other may be preferred. This generic behavior appears disputable in many respects since one overlooks the conclusions expressed by Von Mises which clash against De Finetti’s conceptualization.

The present frame implies that the use of a probability theory is not a matter of opinion and suggests a precise procedure to follow in the professional practice on the basis of incontrovertible facts. Firstly an expert should see whether he is dealing with only one occurrence or with a high number of occurrences. Secondly an expert should follow the appropriate method of calculus depending on the extension (15) or (16) of the subset A.

The present research is still in progress. This paper basically focuses on classical probability and we should extend this study to quantum probability in the next future.

REFERENCES

- D. Gillies, “Philosophical Theories of Probability,” Routledge, London, 2000.
- D. G. Mayo, “Error and the Growth of Experimental Knowledge,” University of Chicago Press, Chicago, 1996.
- M. R. Forster, “How Do Simple Rules ‘Fit to Reality’ in a Complex World?” Minds and Machines, Vol. 9, 1999, pp. 543-564. doi:10.1023/A:1008304819398
- M. Denker, W. A. Woyczyński and B. Ycar, “Introductory Statistics and Random Phenomena: Uncertainty, Complexity and Chaotic Behavior in Engineering and Science,” Springer-Verlag Gmbh, Boston, 1998.
- G. D’Agostini, “Role and Meaning of Subjective Probability; Some Comments on Common Misconceptions,” 20th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, AIP Conference Proceedings, Vol. 568, 2001, 23 p. doi.org/10.1063/1.1381867
- K. R. Popper, “The Logic of Scientific Discovery,” Routledge, New York, 2002.
- T. Tao, “The Strong Law of Large Numbers,” 2012. http://terrytao.wordpress.com/2008/06/18/the-strong-law-of-large-numbers/
- P. Rocchi and L. Gianfagna, “Probabilistic Events and Physical Reality: A Complete Algebra of Probability,” Physics Essays, Vol. 15, No. 3, 2002, pp. 331-118. doi.org/10.4006/1.3025535
- P. Rocchi, “The Structural Theory of Probability,” Kluwer/Plenum, New York, 2003.
- B. Gower, “Scientific Method: A Historical and Philosophical Introduction,” Taylor & Francis, London, 2007.
- L. J. Savage, “The Foundations of Statistics,” Courier Dover Publications, New York, 1972.
- J. Earman, “Aspects of Determinism in Modern Physics,” In: J. Butterfield and J. Earman, Eds., Philosophy of Physics, Part B, North Holland, 2007, pp. 1369-1434. doi.org/10.1016/B978-044451560-5/50017-8
- B. De Finetti, “Theory of Probability: A Critical Introductory Treatment,” John Wiley & Sons, New York, 1975.