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

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 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.

Higher Variations of the Monty Hall Problem (3.0, 4.0) and Empirical Definition of the Phenomenon of Mathematics, in Boole’s Footsteps, as Something the Brain Does

“…What makes Monty Hall 3.0 much more interesting than Monty Hall 1.0 and 2.0 is the following. In Monty Hall 1.0 and 2.0, one always increases one's chances of getting 1 car by switching doors when doors are opened to reveal goats [5]. But in Monty Hall 3.0 and higher, depending on the conditions and what the desired aim is, one's chances of being successful by switching may either decrease or increase.…”

“…In Boole's notation, this coexistence of two facets in a single phenomenon is evoked felicitously by the single symbol admitting of two interpretations. Consider the following two equivalent expressions found in the afore-mentioned article [3]:…”

Chapter Two : Higher Variations of the Monty Hall Problem (3.0, 4.0) and Empirical Definition of the Phenomenon of Mathematics, in Boole��s Footsteps, as Something the Brain Does

“…In Monty Hall 1.0 and 2.0, one always increases one's chances of getting 1 car by switching doors when doors are opened to reveal goats [8]. But in Monty Hall 3.0 and higher, depending on the conditions and what the desired aim is, one's chances of being successful by switching may either decrease or increase.…”

“…In Boole's notation, this coexistence of two facets in a single phenomenon is evoked felicitously by the single symbol × admitting of two interpretations. Consider the following two equivalent expressions found in the afore-mentioned article [6]:…”

“…Or, the 8 sequences that yield no car need to be eliminated. They are sequences 4,8,12,16,20,24,28, and 32. In the list below, the 8 sequences in question have been removed.…”