The Philosophy of Probability Values Behaviour through Fractions and Composite Probability Function for Independent Events in the Discrete Case
Keywords:
Causality, Complement, Experiment, Sequence, Stochastic
Abstract
This paper focuses on dealing with probability theory, and it adopted an objectivism philosophic approach, as well as a mathematical approach. These approaches aim to study the probability values behaviour as discrete quantities in the case of discrete sample space for independent events through fractions and composite functions. This requires a discussion of the usage of probability statements, the causes behind the existence of probabilistic phenomenon, and an explanation that admits to measuring causality in probability theory through the concept of complement and fractions. The paper uses an experiment with a design that addresses the shortcoming of traditional experiments through the concept of fractions. This, in turn, reflects some aspects of the probabilistic behaviour, including some important consequences that follow through. This paper also uses the relative frequency of events and the probability axioms, which provides the sample space to include all possible events in the form of sequences and sub sequences. The paper further provides some definitions that define some elements of the fractions probabilities and the sample space, alongside some proven propositions, lemma, and corollaries that admit to calculating composite probability functions. This is in addition to a brief discussion of the continuous cases.
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References
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34. Montgomery, D. C. (1997). Design and Analysis of Experiments. Fourth Edition. New York, NY: John Wiley & Sons, Inc.
35. Mood, Alexander M., Graybill, Franklin A., & Boes, Duane C. (1974). Introduction to the Theory of Statistics. Third Edition. USA : McGraw-Hill, Inc.
36. Parzen, E. (1962). Stochastic Processes. USA: Holden-Day, Inc.
37. Popper, K. (2002). The Logic of Scientific Discovery. 2nd Edition. Routledge. https://doi.org/10.4324/9780203994627
38. Parzen, E. (1960). Modern Probability Theory and Its Applications. New York, NY: John Wiley & Sons, Inc.
39. Pruss, A. (2013). Probability, Regularity, and Cardinality. Philosophy of Science, 80(2), 231-240. Doi:10.1086/670299.
40. Rényi, A. (1970). Foundations of Probability. San Francisco: Holden-Day.
41. Roberts, F. S. (1979). Measurement Theory. USA: Addison Wesley Publishing Company, Inc.
42. Stoll, M. (1997). Introduction to Real Analysis. Boston: Addison Wesley Longman Inc.
43. Taylor, H. M. & Karlin, S. (1998). An Introduction to Stochastic Modeling. Third Edition. USA: Academic Press.
44. Todhunter, I. (1949). A History of the Mathematical Theory of Probability from the Time of Pascal to Laplace. New York: Chelsea Publishing company.
45. Tucker, H. G. (1967). A Graduate Course in Probability. New York: Academic Press, Inc.
46. Werner, H. (1962). Physics and Philosophy. New York: Harper & Row, Publishers.
2. Athreya, K. B. & Jagers, P. (1997). Classical and Modern Branching Processes: Papers from IMA Workshop Held at the University of Minnesota. Minneapolis, MN, June 13-17, 1994. IMA Volumes in Mathematics and Applications, 84, Springer-Velag, New York.
3. Bartle, R. G. & Sherbert, D. R. (2011). Introduction to Real Analysis. Fourth Edition. USA: John Wiley & Sons, Inc.
4. Bingham, N. H. (2000). Studies in the History of Probability and Statistics XLVI. Measure into Probability: From Lebesgue to Kolmogorov. Biometrika, 87(1), 145–156. http://www.jstor.org/stable/2673568
5. Blanché, R. (1972). L’épistémologie. Paris: Presses Universitaires de France.
6. Borel, E. (1898). Leçons sur La Théorie des Fonctions. Paris: Gauthier-Villars
7. Bohr, N. (1958). Atomic Physics and Human Knowledge. New York: John Wiley and Sons.
8. Brown, H. R. (2011). Curious and Sublime: The Connection between Uncertainty and Probability in physics. Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 369(1956), 4690–4704. http://www.jstor.org/stable/23057214.
9. Brody, B. (Ed.) (1970). Readings in the Philosophy of Science. New Jersey: Prentice - Hall, Inc., Englewood Cliffs.
10. Cantor, G. (1883). Sur Les Ensembles Infinis et Linéaires de Points. ACTA Mathematica 2 (361-371). Paris: A. Hermann.
11. Dunker, T., Li, W. V., & Linde, W. (2000). Small Ball Probabilities of Integrals of Weighted Brownian Motion: Statistics & Probability Letters, 46(3), 0167-7152. Doi:10.1016/S0167-7152(99)00098-X.
12. Dudewicz, E. J. & Mishra, S. N. (1988). Modern Mathematical Statistics. USA: John Wiley & Sons, Inc.
13. Edwards, W, F., Shiflett, R. C., & Shultz, H. S. (2008). Dependent Probability Spaces. The College Mathematics Journal, 39:3, 221-226, Doi: 10.1080/07468342.2008.11922296.
14. Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Volume II. Second Edition. USA: John Wiley & Sons, Inc.
15. Feller, W. (1950). An Introduction to Probability Theory and Its Applications, Volume I. New York, NY: John Wiley & Sons, Inc.
16. Guiraud, P. (1969). La Sémantique. Sixième Édition. France: Presses Universitaires de France.
17. Heol, P. G., Port, S. C., & Stone, C. J. (1971). Introduction to Probability Theory. USA: Houghton Mifflin Company.
18. Heisenberg, W. (1930). The Physical Principles of the Quantum Theory. Chicago: The University of Chicago Press.
19. Hines, W. W. & Montgomery, D. C. (1990). Probability and Statistics in Engineering and Management Science. Third Edition. USA: John Wiley & Sons, Inc.
20. Howson, C. (1995). Theories of Probability. The British Journal for the Philosophy of Science, 46(1), 1–32. http://www.jstor.org/stable/68763.
21. Keynes, J. M. (1921). A Treatise on Probability. London: Macmillan and Co., Limited.
22. Khinchin, A. Ya. (1964). Continued Fractions. Chicago: The University of Chicago Press.
23. Knuth, D. E. (1998). The Art of Computer Programming, Volume II, Third Edition. USA: Addison Wesley.
24. Larson, H. J. (1973). Introduction to the Theory of Statistics. USA: John Wiley & Sons, Inc.
25. Laplace, P. S. (1921). Essai Philosophique sur les Probabilités. Paris: Gauthier-Villars.
26. Lebesgue, H. (1904). Leçons sur L’intégration et La Recherche des Fonctions Primitives. Paris: Gauthier-Villars.
27. Li, W. V. & Shao, Q. -M. (2001). Gaussian Processes: Inequalities, Small Ball Probabilities and Applications. Handbook of Statistics, Volume 19, pages 533-597. Netherlands: Elsevier.
28. Loève, M. (1978). Probability theory, Volume II. 4th Edition. New York: Springer-Verlag.
29. Loève, M. (1977). Probability theory, Volume I. 4th Edition. New York: Springer-Verlag.
30. Marczyk, M. & Wroński, L. (2015). Completion of the Causal Completability Problem. The British Journal for the Philosophy of Science, 66(2), 307–326. http://www.jstor.org/stable/24562936.
31. Meyer, P. L. (1970). Introductory Probability and Statistical Applications. Second Edition. USA: Addison-Wesley Publishing Company, Inc.
32. Merzbacher, E. (1970). Quantum Mechanics. Second Edition. USA: John Wiley & Sons.
33. Mises, R. V. (1957). Probability, Statistics and Truth. Second Revised English Edition. New York: The Macmillan Company.
34. Montgomery, D. C. (1997). Design and Analysis of Experiments. Fourth Edition. New York, NY: John Wiley & Sons, Inc.
35. Mood, Alexander M., Graybill, Franklin A., & Boes, Duane C. (1974). Introduction to the Theory of Statistics. Third Edition. USA : McGraw-Hill, Inc.
36. Parzen, E. (1962). Stochastic Processes. USA: Holden-Day, Inc.
37. Popper, K. (2002). The Logic of Scientific Discovery. 2nd Edition. Routledge. https://doi.org/10.4324/9780203994627
38. Parzen, E. (1960). Modern Probability Theory and Its Applications. New York, NY: John Wiley & Sons, Inc.
39. Pruss, A. (2013). Probability, Regularity, and Cardinality. Philosophy of Science, 80(2), 231-240. Doi:10.1086/670299.
40. Rényi, A. (1970). Foundations of Probability. San Francisco: Holden-Day.
41. Roberts, F. S. (1979). Measurement Theory. USA: Addison Wesley Publishing Company, Inc.
42. Stoll, M. (1997). Introduction to Real Analysis. Boston: Addison Wesley Longman Inc.
43. Taylor, H. M. & Karlin, S. (1998). An Introduction to Stochastic Modeling. Third Edition. USA: Academic Press.
44. Todhunter, I. (1949). A History of the Mathematical Theory of Probability from the Time of Pascal to Laplace. New York: Chelsea Publishing company.
45. Tucker, H. G. (1967). A Graduate Course in Probability. New York: Academic Press, Inc.
46. Werner, H. (1962). Physics and Philosophy. New York: Harper & Row, Publishers.
Published
2023-06-30
How to Cite
Jughaiman, A. (2023). The Philosophy of Probability Values Behaviour through Fractions and Composite Probability Function for Independent Events in the Discrete Case. European Scientific Journal, ESJ, 19(18), 1. https://doi.org/10.19044/esj.2023.v19n18p1
Section
ESJ Natural/Life/Medical Sciences
Copyright (c) 2023 Abdulaziz Jughaiman
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
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