Monty Hall Problem and the Principle of Equal Probability in Measurement Theory
Author(s)
Shiro Ishikawa*
Affiliation(s)
Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan.
Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan.
ABSTRACT
In this paper, we study the principle of equal probability (i.e., unless we have sufficient reason to regard one possible case as more probable than another, we treat them as equally probable) in measurement theory (i.e., the theory of quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics with the Copenhagen interpretation. This turn from physics to language does not only realize theremarkable extensionof quantum mechanicsbut alsoestablish the method of science. Our study will be executed in the easy example of the Monty Hall problem. Although our argument is simple, we believe that it is worth pointing out the fact that the principle of equal probability can be, for the first time, clarified in measurement theory (based on the dualism) and not the conventional statistics (based on Kolmogorov’s probability theory).
In this paper, we study the principle of equal probability (i.e., unless we have sufficient reason to regard one possible case as more probable than another, we treat them as equally probable) in measurement theory (i.e., the theory of quantum mechanical world view), which is characterized as the linguistic turn of quantum mechanics with the Copenhagen interpretation. This turn from physics to language does not only realize theremarkable extensionof quantum mechanicsbut alsoestablish the method of science. Our study will be executed in the easy example of the Monty Hall problem. Although our argument is simple, we believe that it is worth pointing out the fact that the principle of equal probability can be, for the first time, clarified in measurement theory (based on the dualism) and not the conventional statistics (based on Kolmogorov’s probability theory).
Cite this paper
S. Ishikawa, "Monty Hall Problem and the Principle of Equal Probability in Measurement Theory," Applied Mathematics, Vol. 3 No. 7, 2012, pp. 788-794. doi: 10.4236/am.2012.37117.
S. Ishikawa, "Monty Hall Problem and the Principle of Equal Probability in Measurement Theory," Applied Mathematics, Vol. 3 No. 7, 2012, pp. 788-794. doi: 10.4236/am.2012.37117.
References
[1] P. Hoffman, “The Man Who Loved Only Numbers, the story of Paul Erd?s and the Search for Mathematical Truth,” Hyperion, New York, 1998, [2] S. Ishikawa, “Fuzzy Inferences by Algebraic Method,” Fuzzy Sets and Systems, Vol. 87, No. 2, 1997, pp. 181-200. doi:10.1016/S0165-0114(96)00035-8 [3] S. Ishikawa, “A Quantum Mechanical Approach to Fuzzy Theory,” Fuzzy Sets and Systems, Vol. 90, No. 3, 1997, pp. 277-306. doi:10.1016/S0165-0114(96)00114-5 [4] S. Ishikawa, “Statistics in Measurements,” Fuzzy Sets and Systems, Vol. 116, No. 2, 2000, pp. 141-154. doi:10.1016/S0165-0114(98)00280-2 [5] S. Ishikawa, “Mathematical Foundations of Measurement Theory,” Keio University Press Inc., 2006. http://www.keio-up.co.jp/kup/mfomt/ [6] S. Ishikawa, “Monty Hall Problem in Unintentional Random Measurements,” Far East Journal of Dynamical Systems, Vol. 3, No. 2, 2009, pp. 165-181. [7] S. Ishikawa, “A New Interpretation of Quantum Mechanics,” Journal of Quantum Information Science, Vol. 1, No. 2, 2011, pp. 35-42. doi:10.4236/jqis.2011.12005 [8] S. Ishikawa, “Quantum Mechanics and the Philosophy of Language: Reconsideration of Traditional Philosophies,” Journal of Quantum Information Science, Vol. 2, No. 1, 2012, pp. 2-9. doi:10.4236/jqis.2012.21002 [9] S. Ishikawa, “A Measurement Theoretical Foundation of Statistics,” Applied Mathematics, Vol. 3, No. 3, 2012, pp. 183-192. [10] S. Ishikawa, “The Linguistic Interpretation of Quantum Mechanics,” 2012. http://arxiv.org/pdf/1204.3892.pdf [11] S. Ishikawa, “Ergodic Hypothesis and Equilibrium Statistical Mechanics in the Quantum Mechanical World View,” World Journal of Mechanics, Vol. 2, No. 2, 2012, pp. 125-130. doi:10.4236/wjm.2012.22014 [12] S. Ishikawa, “Zeno’s Paradoxes in the Mechanical World View,” 2012. http://arxiv.org/pdf/1205.1290.pdf [13] A. Kolmogorov, “Foundations of the Theory of Probability (Translation),” Chelsea Pub Co. Second Edition, New York, 1960. [14] G. J. Murphy, “C*-Algebras and Operator Theory,” Academic Press, Boston, 1990. [15] J. von Neumann, “Mathematical Foundations of Quantum Mechanics,” Springer Verlag, Berlin, 1932. [16] K. Yosida, “Functional Analysis,” 6th Edition, SpringerVerlag, Berlin, 1980. [17] E. B. Davies, “Quantum Theory of Open Systems,” Academic Press, London, 1976. [18] S. Ishikawa, “Uncertainty Relation in Simultaneous Measurements for Arbitrary Observables,” Reports on Mathematical Physics, Vol. 29, No. 3, 1991, pp. 257-273. doi:10.1016/0034-4877(91)90046-P [19] S. Ishikawa, “What Is Statistics? The Answer by Quantum Language,” arXiv:1207.0407v1 [physics.data-an], 2012. http://arxiv.org/abs/1207.0407v1
[1] P. Hoffman, “The Man Who Loved Only Numbers, the story of Paul Erd?s and the Search for Mathematical Truth,” Hyperion, New York, 1998, [2] S. Ishikawa, “Fuzzy Inferences by Algebraic Method,” Fuzzy Sets and Systems, Vol. 87, No. 2, 1997, pp. 181-200. doi:10.1016/S0165-0114(96)00035-8 [3] S. Ishikawa, “A Quantum Mechanical Approach to Fuzzy Theory,” Fuzzy Sets and Systems, Vol. 90, No. 3, 1997, pp. 277-306. doi:10.1016/S0165-0114(96)00114-5 [4] S. Ishikawa, “Statistics in Measurements,” Fuzzy Sets and Systems, Vol. 116, No. 2, 2000, pp. 141-154. doi:10.1016/S0165-0114(98)00280-2 [5] S. Ishikawa, “Mathematical Foundations of Measurement Theory,” Keio University Press Inc., 2006. http://www.keio-up.co.jp/kup/mfomt/ [6] S. Ishikawa, “Monty Hall Problem in Unintentional Random Measurements,” Far East Journal of Dynamical Systems, Vol. 3, No. 2, 2009, pp. 165-181. [7] S. Ishikawa, “A New Interpretation of Quantum Mechanics,” Journal of Quantum Information Science, Vol. 1, No. 2, 2011, pp. 35-42. doi:10.4236/jqis.2011.12005 [8] S. Ishikawa, “Quantum Mechanics and the Philosophy of Language: Reconsideration of Traditional Philosophies,” Journal of Quantum Information Science, Vol. 2, No. 1, 2012, pp. 2-9. doi:10.4236/jqis.2012.21002 [9] S. Ishikawa, “A Measurement Theoretical Foundation of Statistics,” Applied Mathematics, Vol. 3, No. 3, 2012, pp. 183-192. [10] S. Ishikawa, “The Linguistic Interpretation of Quantum Mechanics,” 2012. http://arxiv.org/pdf/1204.3892.pdf [11] S. Ishikawa, “Ergodic Hypothesis and Equilibrium Statistical Mechanics in the Quantum Mechanical World View,” World Journal of Mechanics, Vol. 2, No. 2, 2012, pp. 125-130. doi:10.4236/wjm.2012.22014 [12] S. Ishikawa, “Zeno’s Paradoxes in the Mechanical World View,” 2012. http://arxiv.org/pdf/1205.1290.pdf [13] A. Kolmogorov, “Foundations of the Theory of Probability (Translation),” Chelsea Pub Co. Second Edition, New York, 1960. [14] G. J. Murphy, “C*-Algebras and Operator Theory,” Academic Press, Boston, 1990. [15] J. von Neumann, “Mathematical Foundations of Quantum Mechanics,” Springer Verlag, Berlin, 1932. [16] K. Yosida, “Functional Analysis,” 6th Edition, SpringerVerlag, Berlin, 1980. [17] E. B. Davies, “Quantum Theory of Open Systems,” Academic Press, London, 1976. [18] S. Ishikawa, “Uncertainty Relation in Simultaneous Measurements for Arbitrary Observables,” Reports on Mathematical Physics, Vol. 29, No. 3, 1991, pp. 257-273. doi:10.1016/0034-4877(91)90046-P [19] S. Ishikawa, “What Is Statistics? The Answer by Quantum Language,” arXiv:1207.0407v1 [physics.data-an], 2012. http://arxiv.org/abs/1207.0407v1