The Monty Hall Problem and beyond: Digital-Mathematical and Cognitive Analysis in Boole’s Algebra, Including an Extension and Generalization to Related Cases

Author(s)
Leo Depuydt

ABSTRACT

The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire monograph has been devoted to its history. There has been a multiplicity of approaches to the problem. These approaches are not necessarily mutually exclusive. The design of the present paper is to add one more approach by analyzing the mathematical structure of the Monty Hall problem in digital terms. The structure of the problem is described as much as possible in the tradition and the spirit—and as much as possible by means of the algebraic conventions—of George Boole’s Investigation of the Laws of Thought (1854), the Magna Charta of the digital age, and of John Venn’s Symbolic Logic (second edition, 1894), which is squarely based on Boole’s Investigation and elucidates it in many ways. The focus is not only on the digital-mathematical structure itself but also on its relation to the presumed digital nature of cognition as expressed in rational thought and language. The digital approach is outlined in part 1. In part 2, the Monty Hall problem is analyzed digitally. To ensure the generality of the digital approach and demonstrate its reliability and productivity, the Monty Hall problem is extended and generalized in parts 3 and 4 to related cases in light of the axioms of probability theory. In the full mapping of the mathematical structure of the Monty Hall problem and any extensions thereof, a digital or non-quantitative skeleton is fleshed out by a quantitative component. The pertinent mathematical equations are developed and presented and illustrated by means of examples.

The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire monograph has been devoted to its history. There has been a multiplicity of approaches to the problem. These approaches are not necessarily mutually exclusive. The design of the present paper is to add one more approach by analyzing the mathematical structure of the Monty Hall problem in digital terms. The structure of the problem is described as much as possible in the tradition and the spirit—and as much as possible by means of the algebraic conventions—of George Boole’s Investigation of the Laws of Thought (1854), the Magna Charta of the digital age, and of John Venn’s Symbolic Logic (second edition, 1894), which is squarely based on Boole’s Investigation and elucidates it in many ways. The focus is not only on the digital-mathematical structure itself but also on its relation to the presumed digital nature of cognition as expressed in rational thought and language. The digital approach is outlined in part 1. In part 2, the Monty Hall problem is analyzed digitally. To ensure the generality of the digital approach and demonstrate its reliability and productivity, the Monty Hall problem is extended and generalized in parts 3 and 4 to related cases in light of the axioms of probability theory. In the full mapping of the mathematical structure of the Monty Hall problem and any extensions thereof, a digital or non-quantitative skeleton is fleshed out by a quantitative component. The pertinent mathematical equations are developed and presented and illustrated by means of examples.

KEYWORDS

Binary Structure, Boolean Algebra, Boolean Operators, Boole’s Algebra, Brain Science, Cognition, Cognitive Science, Digital Mathematics, Electrical Engineering, Linguistics, Logic, Non-Quantitative and Quantitative Mathematics, Monty Hall Problem, Neuroscience, Probability Theory, Rational Thought and Language

Binary Structure, Boolean Algebra, Boolean Operators, Boole’s Algebra, Brain Science, Cognition, Cognitive Science, Digital Mathematics, Electrical Engineering, Linguistics, Logic, Non-Quantitative and Quantitative Mathematics, Monty Hall Problem, Neuroscience, Probability Theory, Rational Thought and Language

Cite this paper

nullL. Depuydt, "The Monty Hall Problem and beyond: Digital-Mathematical and Cognitive Analysis in Boole’s Algebra, Including an Extension and Generalization to Related Cases,"*Advances in Pure Mathematics*, Vol. 1 No. 4, 2011, pp. 136-154. doi: 10.4236/apm.2011.14027.

nullL. Depuydt, "The Monty Hall Problem and beyond: Digital-Mathematical and Cognitive Analysis in Boole’s Algebra, Including an Extension and Generalization to Related Cases,"

References

[1] J. Rosenhouse, “The Monty Hall Problem: The Remarkable Story Behind Math’s Most Contentious Brainteaser,” Oxford University Press, New York and Oxford, 2009.

[2] J. Gill, “Bayesian Methods,” International Encyclopedia of Statistical Science, Springer, Berlin and New York, 2010, pp. 8-10.

[3] R. D. Gill, “The Monty Hall Problem Is not a Probability Puzzle (It’s a Challenge in Mathematical Modelling),” Statistica Neerlandica, Vol. 65, No. 1, 2010, pp. 57-71. doi:10.111/j.1467-9574.2010.00474.x

[4] J. S. Rosenthal, “Monty Hall, Monty Fall, Monty Crawl,” Math Horizons, Vol. 16, No. 1, September Issue, 2008, pp. 5-7.

[5] I. Grattan-Guinness, “The Search for Mathematical Roots,” Princeton University Press, Princeton and Oxford, 2000.

[6] “The symbols for “and” (+) and “or” (–), and for closed-“true” (0) and open-“false” (1) as used in switching algebra, are the inverse of the conventions used in the algebra of logic. However, because of the perfect duality inherent in the algebra, the pos-tulates and theorems are not affected” (W. Keister, A.E. Ritchie, S.H. Washburn, “The Design of Switching Circuits,” The Bell Telephone Laboratories Series, D. Van Nostrand Company, New York, Toronto, and London, 1951, p. 70 note *).

[7] M. Helft, “Google Can Now Say No to ‘Raw Fish Shoes’ in 52 Languages,” The New York Times, March 9, 2010, pp. A1 and A3.

[8] L. Depuydt, “The Other Mathematics: Language and Logic in Egyptian and in General,” Gorgias Press, Piscataway, New Jersey, 2008, pp. 93-95.

[9] J. Venn, “Symbolic Logic,” 2nd Edition, Macmillan and Company, London and New York, 1894, pp. 245-255.

[10] E. Schroeder, “Vorlesungen über die Algebra der Logik (Exakte Logik),” J. C. Hinrichs, Leipzig, Vol. 1, 1890, p. 319.

[11] L. Depuydt, “Zur unausweichlichen Digitalisierung der Sprachbetrachtung: ‘Allein,’ ‘anderer,’ ‘auch,’ ‘einziger,’ ‘(seiner)seits,’ und ‘selbst’ als digitales Wort-feld im ?gyptisch-Koptischen und im Allgemeinen” (“On the Unavoidable Digitalization of Language Analysis: ‘Alone,’ ‘Other,’ ‘Also,’ ‘Only,’ ‘On (his) part,’ and ‘Self’ as a Lexical Field of Digital Purport in Egyptian-Coptic and in General”), to appear in the series Aegyptiaca Monasteriensia as part of the acts of the Workshop “Lexical Fields, Se

[12] L. Depuydt, “The Other Mathematics: Language and Logic in Egyptian and in General,” Piscataway, New Jersey, 2008, pp. 285-306.

[13] Th. Hailperin, “Boole’s Logic and Probability,” North- Holland Publishing Company, New York and Oxford, 1976, p. 131. Also see the 2nd edition, 1986, p. 215.

[1] J. Rosenhouse, “The Monty Hall Problem: The Remarkable Story Behind Math’s Most Contentious Brainteaser,” Oxford University Press, New York and Oxford, 2009.

[2] J. Gill, “Bayesian Methods,” International Encyclopedia of Statistical Science, Springer, Berlin and New York, 2010, pp. 8-10.

[3] R. D. Gill, “The Monty Hall Problem Is not a Probability Puzzle (It’s a Challenge in Mathematical Modelling),” Statistica Neerlandica, Vol. 65, No. 1, 2010, pp. 57-71. doi:10.111/j.1467-9574.2010.00474.x

[4] J. S. Rosenthal, “Monty Hall, Monty Fall, Monty Crawl,” Math Horizons, Vol. 16, No. 1, September Issue, 2008, pp. 5-7.

[5] I. Grattan-Guinness, “The Search for Mathematical Roots,” Princeton University Press, Princeton and Oxford, 2000.

[6] “The symbols for “and” (+) and “or” (–), and for closed-“true” (0) and open-“false” (1) as used in switching algebra, are the inverse of the conventions used in the algebra of logic. However, because of the perfect duality inherent in the algebra, the pos-tulates and theorems are not affected” (W. Keister, A.E. Ritchie, S.H. Washburn, “The Design of Switching Circuits,” The Bell Telephone Laboratories Series, D. Van Nostrand Company, New York, Toronto, and London, 1951, p. 70 note *).

[7] M. Helft, “Google Can Now Say No to ‘Raw Fish Shoes’ in 52 Languages,” The New York Times, March 9, 2010, pp. A1 and A3.

[8] L. Depuydt, “The Other Mathematics: Language and Logic in Egyptian and in General,” Gorgias Press, Piscataway, New Jersey, 2008, pp. 93-95.

[9] J. Venn, “Symbolic Logic,” 2nd Edition, Macmillan and Company, London and New York, 1894, pp. 245-255.

[10] E. Schroeder, “Vorlesungen über die Algebra der Logik (Exakte Logik),” J. C. Hinrichs, Leipzig, Vol. 1, 1890, p. 319.

[11] L. Depuydt, “Zur unausweichlichen Digitalisierung der Sprachbetrachtung: ‘Allein,’ ‘anderer,’ ‘auch,’ ‘einziger,’ ‘(seiner)seits,’ und ‘selbst’ als digitales Wort-feld im ?gyptisch-Koptischen und im Allgemeinen” (“On the Unavoidable Digitalization of Language Analysis: ‘Alone,’ ‘Other,’ ‘Also,’ ‘Only,’ ‘On (his) part,’ and ‘Self’ as a Lexical Field of Digital Purport in Egyptian-Coptic and in General”), to appear in the series Aegyptiaca Monasteriensia as part of the acts of the Workshop “Lexical Fields, Se

[12] L. Depuydt, “The Other Mathematics: Language and Logic in Egyptian and in General,” Piscataway, New Jersey, 2008, pp. 285-306.

[13] Th. Hailperin, “Boole’s Logic and Probability,” North- Holland Publishing Company, New York and Oxford, 1976, p. 131. Also see the 2nd edition, 1986, p. 215.