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Advances in Pure Mathematics, 2011, 1, 136-154 doi:10.4236/apm.2011.14027 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM The Monty Hall Problem and beyond: Digital-Mathematical and Cognitive Analysis in Boole’s Algebra, Including an Extension and Generalization to Related Cases Leo Depuydt Department of Egy pt olo gy a n d Ancient W est ern Asi a n St u d i es , Brown University, Providence, Rhode Island, USA E-mail: Leo_Depuydt@brown.edu Received April 27, 2011; revised May 14, 2011; accepted May 25, 2011 Abstract The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire mono- graph has been devoted to its history. There has been a multiplicity of approaches to the problem. These ap- proaches 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 ana- lyzed digitally. To ensure the generality of the digital approach and demonstrate its reliability and productiv- ity, 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. Keywords: Binary Structure, Boolean Algebra, Boolean Operators, Boole’s Algebra, Brain Science, Cognition, Cognitive Science, Digital Mathematics, Electrical Engineering, Linguistics, Logic, Monty Hall Problem, Neuroscience, Non-quantitative and Quantitative Mathematics, Probability Theory, Rational Thought and Language 1. Introduction The Monty Hall problem, named after the television host Monty Hall who made it famous in a TV show, has re- ceived its fair share of attention in mathematics. Recently, accessibility to the history of the problem was greatly enhanced due to the appearance of a monograph devoted entirely to the subject [1]. A multiplicity of approaches has been applied to the problem. These approaches are not necessarily mutually exclusive. Among key contribu- tions of most recent date to the problem’s analysis are an updated statement of the Bayesian analysis of the prob- lem [2], a challenge to move towards a mathematical modeling of the problem [3], and yet other innovative treatments [4]. The design of the presen t paper is to add one more ap- proach 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 th e tradition and the spirit, and 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 [5]. This paper has four main parts. The digital approach is L. DEPUYDT 137 ppp outlined in general in part 1. In part 2, the Monty Hall problem is analyzed digitally. The Monty Hall problem involves 3 doors, 1 car, 2 goats, 1 picked door, and 1 opened door. To ensure the generality of the digital ap- proach and to demonstrate its reliab ility an d produ ctiv ity, it would seem to be critically important to extend and generalize the analysis of the Monty Hall problem to any number of doors, cars, opened doors, and picked doors in light of the axioms of probability theory. Such an exten- sion and generalization is the subject of parts 3 and 4 of this paper. Part 3 concerns an extension to any number of doors, cars, opened doors, and picked doors. Part 4 con- cerns an additional extension to any number of picked doors. The pertinent mathematical Equations are devel- oped and presented and illustrated by means of exam- ples. 2. Preliminary Considerations and Reflections on Digitality and Cognition 2.1. Digital Mathematics and Quantitative Mathematics Digital mathematics is the mathematics in which nothing gets bigger or smaller and everything is either On or Off, 1 or 0. Different notation systems are prevalent in dig ital mathematics. I prefer the notation used by the Father of the digital age, George Boole. Digital mathematics needs to be differentiated from the search for the roots of mathematics, a subject to which Bertrand Russell and Kurt Gödel and many others have contributed. These efforts are often called logistics, as opposed to logic. A recent voluminous book on the history of logistics from 1870 to 1940 documents all these contributions in detail. This account also reveals that it seems to be hardly the case that the ultimate roots of mathematics have been once and for all fully uncov- ered [5]. At the outset of his Elements of Algebra (Vollständige Anleitung zur Algebra), Euler states that mathematics is the science of quantity, the systematic study of that which is capable of increase or diminution. This state- ment is not fully complete. Digital mathematics is fun- damentally different from the mathematics with which one is better familiar. In digital mathematics, nothing gets bigger or smaller. When one performs a Boolean search on the Internet looking for all that is Paris and in addition all that is Paris–adding Paris to Paris as it were by using the so-called Boolean OR-operator—the information that one gets is not in any way larger than if one had just searched for Paris alone. Adding Paris to Paris does not produce a class or set that is twice as large as Paris. If p is Paris, then 2pp p in Boole’s algebra. By contrast, in the familiar mathematics, quantity mathematics, . Furthermore, when one executes a Boolean search for all entities that hav e two properties, namely b eing French and again being French—multiplying French by French as it were by using the so-called Boolean AND-opera- tor—one does not obtain search information th at is larger than if one had just searched for all that is French . If f is French, then in Boole’s algebra, f ff 2 . By contrast, in the more familiar mathematics, quantity mathematics, f ff. Something is getting bigger. 2.2. Boole’s Algebra and the Algebra of Electrical Engineering It should be noted that, when Boole’s algebra was adopted in electrical engineering, the conventions were switched. Boole’s “0” is electrical engineering’s “1” and vice versa. Boole’s “×” is electrical engineering’s “+” and vice versa [6]. I assume that Claude Shannon, who adapted Boole’s algebra for the design of switching cir- cuits in the 1930s and thus in a sense became the Father of computer science, was the originator of this change. In electrical engineering, 0 is conceived as zero resistance or hindrance and therefore as an open circuit. I person- ally prefer Boole’s notation and will use it in what fol- lows. But as long as it is understood which functions symbols have, it makes no difference whether one or the other notation is used. 2.3. The Digital Nature of Rational Thought and Language The human experience consists entirely of how the brain engages reality outside itself by means of the senses, nothing more, nothing less. This includes any manner in which the brain combines sensory perceptions internally. There are more than the traditional five senses, sight, hearing, smell, taste, and touch. The others include sens- ing pain, sensing that one is upside down, sensing resis- tance when pushing, and sensing hunger. Part of the brain’s engagement with what is outside itself may be called rational thought and language—as distinct from, say, emotions. I refrain from defining at this point exactly what is ra- tional thought and what is language in rational thought and language. It is much preferable to begin by regarding the two together as a single large pheno menon. It may in fact be difficult to disentangle the two entirely. After all, to which of the two do any connections between the two belong? I am personally convinced that rational thought and Copyright © 2011 SciRes. APM L. DEPUYDT 138 1c language is entirely digital. This fun damental assumptio n will serve as a working hypothesis. I have begun con- structing a comprehensive model of how rational thought and language proceeds digitally. But the presentation of this model is reserved for another occasion. In seeking inspiration as to what this model might look like in the brain, there is nothing wrong with creating physical tem- plates consisting of magnetic coils and switches or tran- sistors or memristors in an attempt to create a prefigura- tion of what will be found later in th e brain. The proposed model has nothing to do with any of the many programs now in ex istence that allow a machine to comprehend, produce, or translate languages. These pro- grams are impressive, as appears from a translation by Google Translate of a passage of Antoine de Saint- Exupéry’s Le petit prince presented in a recent article in the New York Times [7]. These programs do not, how- ever, in any way mimic human language nor do they pretend to. All are based on probability and statistics. Relying on huge databases and much computing power, the programs mathematically predict what is most likely to come next based on information already stored. These intelligent guesses are fast beco ming ever more accurate. 2.4. Empirical Basis for Observing the Digitality of Rational Thought and Language Little or nothing is known directly about how the bio- logical brain produces thought and language. The ques- tion arises: Is it not premature to construct models per- taining to how the brain thinks and talks? Where is the empirical basis? The empirical basis is twofold. First, it is abundantly clear that the brain teems with digital activity, even if the precise mechanisms of this activity are mostly not understood. Second, as one brain communicates with another through thought and language, all communica- tions need to travel by air from the mouth of a speaker to the ear of a hearer or by light from the written page to a reader’s eyes. There can be no doubt that ever ything that is essential to the structure of rational thought and lan- guage must be conveyed in sound waves or light beams that travel from mouth to ear or from page to eye. One might object that sounds and written symbols are not the same as operations of neurons inside the brain. Then again, certain operations of neurons generate a structure that is empirically accessible in language. If the structure of the neurons differed from the linguistic structure they spawn, people would say things that differ from what they think that they are saying. Clearly, the structure expressed in language must be exactly the same as the structure formed inside the brain, even if the material platforms that the two inhabit could hardly dif- fer more. What about the validity of the proposed digital model? It is true that mathematical models have predictive value. Consider the computations relating to a novel kind of bridge construction. The computations are predictive in the sense that, if they are error free, the bridge should not collapse. What is more, the computations are binding. The bridge must be built according to the computations or it will collapse. The validity of digital mathematics in g eneral h as been amply demonstrated by countless applications in tele- phone circuits and computer science. Still, one cannot conjure up just any fanciful digital analysis of brain op- erations and simply expect the brain to operate according to it. The digital analysis must meet certain empirical conditions and be comprehensive in the mathematical sense by extending to all possible cases. The digital analysis should be to linguistic reality what mathematics is to physical reality in the field of physics. 2.5. The Digital Supplements When one looks at a page of written text, the 1s and the 0s do not readily jump at the eye. So where is the digital structure? In a course that I might one day teach about the digital nature of rational thought and language, I might begin by confronting students with the expression “two black cats” and ask where the mathematics is in this expression. I would suspect that quite a few might point to “two” as the mathematical component. However, “black cats” is just as mathematical. In digital terms, the presence of something creates a certain awareness of its absence, in other words, of all that it is not. Accordingly, to the class or set of cats cor- responds a supplement class or supplement of all that is not cats. If the class of cats is denoted by c, then its su p- plement is denoted in Boole’s notation by , that is, the universe or all that one could possibly think about (Boole’s “1”) minus (–) cats (c), or also by c. Likewise, the class of all that is black can be denoted by b and its supplement, all that is not black, by b bc . 2.6. The Digital Combination Classes Furthermore, digitally speaking, two classes “black” (b) and “cats” (c), along with their respective supplements, divide the universe or all that one could possibly think about into exactly four combination classes, black cats, non-black cats, what is black but not a cat, and what is neither black nor a cat. The present writer belongs to th e fourth category. In Boole’s notation, the combination classes are denoted by (or bc), bc (or bc ), bc (or bc ), and bc (or bc ). The universe or all Copyright © 2011 SciRes. APM L. DEPUYDT 139 that one could possibly think about, Boole’s “1,” consists entirely of the sum of the four combination classes, as follows. 1 (or also:1 bcbc bc ) bcbc bcbc bc (1) This division is profoundly digital. It encompasses all the combinations in which presence and absence, or On and Off, or 1 and 0, of two classes “cat” and “black” and their supplements can combine. The division is also fundamental to how we think. Consider the simple sentence “The cat is black.” The two classes “the cat” and what is black generate four combi- nation classes. What matters is the abolition of one of the four combination classes, namely what is both the cat and not black. It is this operation of abolition that makes the thought “The cat is black” possible. In Boole’s algebra, there are two levels of thought, the primary level of th e things and the secondary level of the events. A proposition such as “The sun shines” is pri- mary. The secondary level can be comprised of two pri- mary propositions. An example is “When the sun shines, I take a walk on the beach.” In this sentence, one digital combination class is switched off or shut down or empty, that is, all the occasions when the sun shines and I do not take a walk on the beach. There is no such thing accord- ing to said statement. The three other digital co mbination classes are on or occupied: either the sun shines and I walk on the beach, or the sun does not shine and I walk on the beach, or the sun does not shine and I do not walk on the beach. The relation to the conditio sine qua non can be ex- plained digitally [8]. Compare the statement already mentioned, “When the sun shines, I take a walk on the beach,” with the statement “Only when the sun shines do I take a walk on the beach.” A different digital combina- tion class is switched off in the latter statement, namely all the occasions when the sun does not shine and I do take a walk on the beach. Furthermore, in the sentence “I take a walk on the beach if and only if the sun shines,” both said combina- tion classes are switched off. Since there are four digital combination classes, that means that two combination classes remain switched on: either I walk on the beach and the sun shines or I do not walk on the beach and the sun does not shine. 2.7. Limits to the Universe When two classes and their supplements partition the universe or all that one could possibly think about into four combination classes, it is common to impose un- spoken or explicit limits on what is being partitioned [9]. One hardly always considers everything thinkable. For example, the statement “Manchester is the winner” de- scribing the outcome of a soccer match between Man- chester United and Liverpool involves four digital com- bination classes: all that is Manchester and the winner, all that is Manchester but not the winner, all that is not Manchester yet the winner, and all that is neither Man- chester nor the winner. The design of the statement “Manchester is the winner” is to present two combina- tion classes as empty: all that is Manchester but not the winner and all that is not Manchester yet the winner. It seems clear that all that is not Manchester does not in this case include the Queen of England or the Pope in Rome. Non-Manchester is limited to the soccer club Liverpool. It is also clear that “Manchester” does not refer to all of the city of Manchester, but just to the soc- cer club Manchester United. Although the limits i mposed on the universe are not stated explicitly, it seems clear what they are. 2.8. Excursus: On Negation and on the Digitality in Rational Thought and Language In a digital world, negation is the mother of all meanings. Everything without exception can be negated: “Caesar” as “not Caesar,” “It rains” as “It does not rain,” “there” as “not there,” “There is” as “There ain’t,” and “yes” as “no.” It is as if reality presents itself to us in two parallel universes, the affirmative and the negated. But what is negation? E. Schröder, the mathematician and onetime director of the Technische Hochschule in Karlsruhe, advised great caution when it comes to defining negation because the most famous philosophers from Aristotle to Kant proposed definitions of negation that are very far apart and great authorities constructed untenable theories about negation that exhibit the greatest internal contra- dictions [10]. Greek philosophers struggled mightily with being and not being and being and b ecoming and th e like. But it is only since the mid-nineteenth century that digi- tal mathematics has provided what I believe to be the valid and final definition of negation. What is negation in digital terms? As the brain engages reality outside itself as observed or as remembered or even as recombined, it naturally does not focus on, or contemplate, everything all at once. It selects certain components of what Boole calls the universe, that is, all that one could possibly think about. To a certain degree, by the way it is structured, reality presumably more easily draws the attention of the brain to certain of its facets rather than to others. One may Copyright © 2011 SciRes. APM 140 1 L. DEPUYDT focus on an entity such as a tree, on a property that comes with that entity such as green, or on a circum- stance in which that entity along with its property finds itself such as “in the forest.” The contemplation of any component of all that one could possibly think about automatically conjures up the notion of all that remains, all that that component is not, for example, all that is not a tree, all that is not green, or all that is not in the forest. If the component of the universe of thought is viewed as a class or set, then all that is not that component is just as much a class. A class that encompasses all that some- thing is not may be called a supplement class or a sup- plement. As was noted above, in Boole’s algebra, if lower case g denotes all that is green, then all that is not green is denoted by g , that is, the universe (Boole’s “1”) minus g, which Boole abbreviates as g . Negation is born, and so is digitality, when the need arises to refer explicitly to all that something is not. For example, I may want to state that a certain car falls out- side the class of all that is green. I can do so by means of the word “not,” as in “This car is not green.” Importantly, a class and its supplement together make up all that is thinkable. In this connection, Aristotle for- mulated the fundamental axiom of thought, namely that something cannot at the same time be and not be some- thing. What is more, the relation between a class and its supplement is like a toggle. The original class is of course all that its supplement class is not. Put differently, the original class is the supplement of its own supple- ment. One common way of referring to this property of digital reality is that two negations cancel one another. This is not the place to illustrate the digital component of rational thought and language at length. Suffice it to point to a semantic field among whose members are English words such as “alone,” “also,” “only,” “other,” and “self” and an English expression such as “for his part” along with their equivalents in other languages [11]. Everyone knows how to use these words. But defining them is another matter. Without entering into detail, it would appear that these words all refer to digital sup- plement classes. For example, “he alone” means “no others besides him,” “no non-he’s” as it were. “He also” means “others besides him,” that is, “non-he’s” in addi- tion to him. In a digital world, there is also a need for referring with an explicit word to the supplement class. The word “other” performs exactly that function. “Other” refers to what something else that has been men- tioned is not. In this connection, I have also proposed to analyze contrastive emphasis digitally [12]. When one says, for example, “It is in Paris that the session is held” or “The session is held in Paris,” one apparently means “not somewhere else,” that is, “not in non-Paris.” Paris is presented as the digital supplement of its digital supple- ment, which is very much Paris itself. 3. Two Goats and a Car: Digital-Mathematical Analysis of the Monty Hall Problem 3.1. Boole and Probability It is now generally forgotten that Boole wrote the Magna Charta of the digital age, his Investigation of the Laws of Thought (1854), to address problems in probability. But his contribution to probability has been “simply bypassed by the history of the subject” [13]. Boole believed that the theory of probability is a field of mathematics strad- dling the fence that separates quantitative mathematics from digital mathematics. It is assumed here as a working hypothesis that the digital approach permeates all engagement of the brain with reality, and that includes assessments of probability. In support of this assumption, the Monty Hall pro blem is analyzed digitally in the present part 2 and then extended and generalized in ligh t of the axioms of probability the- ory in parts 3 and 4. The study of the Monty Hall problem has a long his- tory. But the countless technical and popular treatments of the problem are characterized by the exclusion of a potentially fertile additional approach, the digital and Boolean perspective, the perspective that I personally believe reflects the fundamentally digital nature of how the brain processes reality in terms of rational thought and language. 3.2. Description of the Monty Hall Problem Behind 3 closed doors, 2 goats and 1 car are hiding. One is asked to pick a door to get what is behind it. The aim is to get the car. One begins by picking a closed door without however knowing or being told what it is hiding . Subsequently, someone who knows what is behind all 3 doors without revealing this knowledge to the person who has picked a closed door opens 1 of the 2 doors that were not picked, and more specifically a door hiding a goat. Since 2 of the 3 doors hide goats, it is always pos- sible to open a non-picked door that hides a goat. The other 2 doors remain closed and 1 of these 2 is the one that was initially picked. The Monty Hall problem revolves entirely around the following question. Once a door has been opened to re- veal a goat, should one switch from the closed door that one has picked to the other door that remains closed in Copyright © 2011 SciRes. APM L. DEPUYDT 141 order to improve one’s chances of getting the car? There is no doubt whatsoever that one should switch. The need is therefore for establishing the respective probabilities of switching and not switching doors. 3.3. Intuitive Grasp of the Need to Switch Doors: Opening Doors to Reveal Goats as a Means of Compressing or Condensing a Probability into Fewer Doors The need for switching doors to improve one’s chances of getting the car is obvious from the following consid- eration. Everyone knows that one has only 1 chance in 3 of getting the car by picking a certain door. That in effect means that there are 2 chances in 3 that the car is hiding behind 1 of the 2 other doors that were not picked. In other words, there are 2 chances in 3 that the 2 non-picked doors hide 1 car and 1 goat. Accordingly, there ought to be every temptation to switch to the other 2 doors to get the car. The problem is that one cannot switch to both of the 2 other doors that were initially not picked. One can only switch to 1 of them. But to which one of the 2 should one switch? This is where critical and odds-changing help arrives in the form of someone opening 1 door to reveal 1 goat. The effect of this intervention is that the chance of 2 in 3 that the 2 other doors are hiding the car is concentrated in 1 unopened door. What opening 1 door to reveal 1 goat achieves is to compress or condense a certain degree of probability distributed equally over a number of doors into fewer doors. In the case of the Monty Hall p roblem, a probability of 23, which is distributed equally as a probability of 13 i C over each of the 2 doors that are ini- tially not picked, is compressed into 1 door by opening 1 door to reveal 1 goat. In sum, there is every reason to switch doors. It is not certain that one will get the car. Bu t one has 2 chan ces in 3 of being lucky. 3.4. The Two Digital Levels, the Level of Things and the Level of Events Just as two classes “black” and “cat” digitally generate four combination classes (§2.6), the Monty Hall problem in a digital and Boolean perspective fundamentally in- volves two classes generating four combination classes. But in the case of the Monty Hall problem, the two classes do not contain all the instances of two thin gs, such as “black things” and “cats,” but rather all occa- sions or instances or occurrences of two events. Accord- ing to Boole’s analysis, rational thought and language exhibits two levels, the level of primary proposition s that is concerned with classes of things and the level of sec- ondary propositions that is concerned with classes of events. Each class of events contains all the occurrences of a certain event. When two classes of events and their supplement classes are partitioned into four combination classes, it is common to impo se un spok en o r explicit li mits o n what is being partitioned (§2.7). The same applies to the Monty Hall problem and any extensions thereof. What one is considering in terms of things is limited to a number of doors hiding either cars or goats. What one is consider- ing in terms of events is limited to initially pickin g or not picking a car and then picking or not picking a car by switching after one or more doors have been opened to reveal goats. 3.5. Mathematical Notation of Things and Events To differentiate things and events in mathematical nota- tion, things are denoted below by lower case italic letters and events by upper case italics. For example, c stands for “cars” and C for picking a door hiding a car. In addition, subscript letters provide additional distinctive information about events. For example, will stand for initially picking a door hiding a car. 3.6. The Two (Classes of) Events Involved in the Monty Hall Problem, the Second Dependent on the First In the case of the Monty Hall Problem, two events are involved. What is more, the two events occur in a fixed sequence. The second event always follows the first and is dependent upon the first according to the definition of dependence in the classic theory of probability. The two classes of events are as follows. The first contains all the occasions when one initially picks the door behind which the car is hiding. The second contains all the occasions when one picks the door behind which the car is hiding by switching from the door initially picked to the sole door that remains closed after a door has been opened to reveal a goat. 3.7. The Supplement Classes, or Supplements, of the Two Events Involved Digitally speaking, all classes come with supplement classes or supplements. A supplement contains all that a class is not. A supplement is a class in its own right. On the level of things, the supplement of “cat” is all that is not a cat. On the level of events, the supplement of “It rains” is all the occasions when it does not rain. In the case of the Monty Hall problem, the supplement Copyright © 2011 SciRes. APM 142 C L. DEPUYDT of initially picking the door hiding the car contains all the occasions when one fails to initially pick that door. The supplement of picking the door hiding the car by switching from the door initially picked to the sole other door remaining closed after a door has been opened to reveal a goat contains all the occasions when one fails to pick the door hiding the car by switching doors. 3.8. The Two Classes and Their Supplements in Boole’s Algebra In Boole’s algebra, picking the door hiding the car may be denoted by and failing to pick that door byC 1 i C . Boole uses overstrike as an abbreviation of 1–, as in , that is, the universe or any events that one could possibly think about minus (–) all the occasions when the door hiding the car is picked. C Initially (i) picking the door hiding the car may be de- noted by and failing to initially pick that door by i C. Furthermore, picking the door hiding the car by switching (s) doors once a door has been opened to re- veal a goat may be denoted by s Cand failing to pick that door by switching doors by s C i C . Since initially picking a car () is the same as ini- tially not picking a goat (i G), and so on, the following Equations apply. ii CG ii CG s s CG s s CG CC 3.9. The Four Digital Combination (Classes of) Events In digital fashion, two classes of events along with their supplements generate four combination classes corre- sponding to all the four possible combinations of the occurrences and non-occurrences of the two events. The four combination events are as follows: (1) (On-On) initially pick the car and then again pick the car by switching doors; (2) (On-Off) initially pick the car but then fail to pick the car by switching doors; (3) (Off-On) initially fail to pick the car but then suc- ceed in picking the car by switching doors; and (4) (Off-Off) initially fail to pick the car and then again fail to pick the car by switching doors. Digital combination class (1) is denoted in Boole’s notation by is or as is CC ;is is yet another notation. Boole’s “×”—now better known as the Boolean AND-operator even if Boole himself did not refer to it in this—manner denotes a combined event in which two component events are valid at the same time. The other three combination events may be denoted by CC s i CC , is CC , and s i CC C . 3.10. The Probabilities of the Digital Combination Events In Boole’s algebra, four symbols such as i, i, C s C, and s C CC not only denote classes of events but also the probability that the events in question will take place. Accordingly, the probabilities of the four digital combi- nation events (1), (2), (3), and (4) listed in §3.9 consist for each combination event of the product of the prob- ability that the first of the two combined events will take place multiplied by the probability that the second event will. The four probabilities may therefore be denoted by is , s i CC ,is CC , and s i respectively. The symbol “×” is then to be understood quantitatively and not as an equivalent of what is now known as the Boolean AN D- operator. CC C 3.11. 100% as the Sum of the Probabilities of All Combination Events The two classes “cat” and “anything black” along with their supplements subdivide all that is thinkable in terms of things, as denoted by Equation (1) in §2.6. Likewise, the two classes i and s C, along with their supple- ments subdivide all that is thinkable in terms of events, as denoted by Equat i on (2). 1isisis CC CC CC is CC C (2) In other words, the totality of all possible scenarios consists entirely of four combination events: either i and s C C both happen, or i does and s C does not, or i does not but C s C does, or neither i nor C s C i C do. Put differently, either and s C co-occur, or i C and C s do, or i and C s C do, or and i C s C do. There are no other possibilities. Accordingly, the probabilities of the four combination events add up to 1 or 100%. It is one hundred percent certain that one of the four combination events will take place. 3.12. The Two Empty Digital Combination Classes of Events Closer reflection reveals that two of the four combination events involved in the Monty Hall problem never occur, namely is CC and s i CC . The corresponding classes of events are therefore empty. Digitally speaking, these two combination events are switched off, as it were. Copyright © 2011 SciRes. APM L. DEPUYDT Copyright © 2011 SciRes. APM 143 CC As regards is , one cannot initially pick the car and then again pick the car by switching doors because there is only 1 car. As regards s i, one cannot ini- tially fail to pick the car—in other words, pick a goat— and then again fail to pick the car—that is, again pick a CC goat—by switching doors. The reason is that, when one has initially picked a goat, the door hiding the only other goat is opened and one cannot switch to that door be- cause it is now open. Accordingly, the only door to which one can switch hides the car and one cannot fail to pick the car by switching doors. The probability of these two combination events is therefore equal to zero (0). Equation (2) in §3.11, which denotes the sum of the probabilities of the four digital combination events, can therefore be reduced to a sum of two combination events, as follows. i s is CC CC1 (3) 3.13. The Digital Combination Event Including the Desired Outcome of Getting the Car (C), namely is CC C The desired outcome of the Monty Hall problem as a challenge is getting the car (). Of the two digital com- bination classes of events that are not empty in Equation (2) in §3.12, namely is and CCis CC, only is CC includes the desired outcome C. Importantly, this com- bination event involves switching doors, as denoted by subscript s, after initially failing to pick the car (i). In the other combination event, one had initially picked the car , but then loses it by switching doors. C Ci If the probability of the combination event with the desired outcome is lower than 0.5 or 50%, one should not switch doors to increase one’s chances of getting the car. If the probability is higher than 0.5 or 50%, switch- ing doors makes getting the car more probable. If the probability is exactly 0.5 or 50%, it does not make a dif- ference whether one switches doors or not; one does not increase or decrease one’s chances of getting the car. What is the probability of the combination event that includes the desired outcome? Two of the four digital combination classes of events are empty (§3.12). Accordingly, the sum of the prob- abilities of the two other combination events is 1 or 100%. Either one or the other of the two other combina- tion events must be the case. If the probability of one of the two combination events is x%, then the probability of the other is (1 3.14. The Probability of the Digital Combination Event Including the Desired Outcome, namely x )% be- cause there are only two. The probability of one combi- nation event can be derived from the probability of the other by subtracting the probability of the other from 1 or 100%. is CC The desired outcome is achieved by switching doors as part of the digital combination event is . In order to compute the probability of this combination event, it is first necessary to establish the probabilities of its two components, the two events CC i C and s C. The probability of i C, that is, failing to initially pick the car, is evidently 23 or about 66.7%. There are 3 doors and 2 of them hide a goat. C In establishing the probability of s , it needs to be taken into account that s C is a dependent event ac- cording to the definition of dependence in classical probability theory. Accordingly, the question to ask is: What is the probability that one will pick the car by switching ( s C) after having initially failed to pick it (i C)? That probability is 1 or 100%. Indeed, if one first picked a goat and the other door hiding a goat is opened, switching to the only other unopened door must always result in picking the car. The probability of the desired combination event i × C s C, which results in picking the car by switching doors ( s C), is therefore 23 123 . Not only does this combination event produce the desired outcome but its probability is also higher than 0.5 or 50%. There can be no doubt: One has to switch doors to improve one’s chances of getting the car. 3.15. The Probability of the Combination Event That Fails to Achieve the Desired Outcome, namely s i CC Failing to get the car is the outcome of the digital com- bination event i C s C . After initially picking the car (i), one loses it by switching doors (C C s ). The prob- ability of this digital combination event can be derived directly from the probability of the other digital combi- nation event is CC (§3.14). That is because only two of four possible digital combination events actually oc- cur in the Monty Hall problem. Their combined prob- ability must therefore be 1 or 100%. Either one or the other must take place. Since the probability of is CC is 23 (§3.14), the probability of s i C must be C 123 or 13. But for completeness’ sake, it may be desirable to establish the probability of s i CC in its own right. The probability of initially picking the car (i) is C13 or about 33.3%. There is 1 car and there are 3 door s. But once the car is chosen, the probability of failing to pick the car by switching doors ( C s ) is 1 or 100% because one has already picked the sole car and cannot pick it 144 L. DEPUYDT again. Consequently, the probability of s i CC is 131 or 13 . 4. Extension and Generalization of the Monty Hall Problem to Any Number of Doors (d), Cars (c), and Opened Doors (o) 4.1. The Probability of Getting a Car by Switching Doors (Cs): cd- - -o 1 1 dd The general question that stands at the center of the lar- ger problem of which the Monty Hall problem represents just one specific case is as follows: Should one switch doors in order to improve one’s chances of getting cars after doors have been opened to reveal goats? To answer the question, one needs to know the respective probabili- ties of picking a door hiding a car by switching doors ( s C) and picking a door hiding a car by not switching. The probability of picking a door hiding a car by not switching is the same as the probability of initially pick- ing a door hiding a car (i C). If one does nothing else, the probability does not change. If the probability of getting a car by switching is greater than the probability of get- ting a car by not switching, then one should switch doors to improve one’s chances of getting a car. A comparison between the probabilities of s Cand imposes itself. i Of the two probabilities in question, that of picking a door hiding a car by not switching is readily known. As was said, it is the same as the probability of initially picking a door hiding a car (i C). This probability is the ratio of cars to doors and can be denoted by C cd. It is the ratio of favorable outcomes to the sum of favorable and unfavorable outcomes, in accordance with a funda- mental axiom of classical probability theory. For exam- ple, if there is 1 car and there are 3 doors, the chance of initially picking a door hiding a car is 13 and 13 remains the chance of getting the car if one does nothing else, such as switching to another door. If there are 2 cars and 5 doors, the chance of getting a car by sticking with the door that was initially picked is 25. And so on. As regards the other probability, that of getting a car by switching doors ( s C), Equation (4) applies. 1 1 cd dd o s C (4) c = the number of cars (= g , the number of non-goats) d = the number of doors (=, the sum of cars and goats) cg 1p1p dcg cdg o = the number of opened doors How Equation (4) is obtained is explained below in §§4.4-6. In Equation (4), 1 is the number of picked doors (p). The picked doors are not denoted algebraically by, say, p because Equation (4) is only valid when there is only 1 picked door, that is, when . Cases in which there is more than 1 picked door, that is, in which , are discussed in part 5 bel o w. Equation (4) does not feature goats (g). However, of the three variables d, c, and g, each can be derived from the two others because , , and g dc . When the number of the cars is 1, Equation (4) can be simplified as follows. scg g Cdg odg o (5) After all, (4) is equivalent to the following Equation . 1 1 s cc g Cdc go (6) 1c If , then the following Equation can be substi- tuted for (6). 11 1 11 s g Cdg o (7) And Equation (7) can be rewritten as Equation (5). As will be seen below, the deeper reason for this simplifica- tion is as follows. When there is more than 1 car, getting a car by switching doors can be the result of both initially picking a door hiding a car and initially picking a door hiding a goat. There are two scenarios. When there is only 1 car, getting the car by switching doors can only be the result of initially picking a door hiding a goat. There is only one scenario. One cannot initially pick the only car and then pick it again when switching doors. d-d- -o11 4.2. as the Factor by Which the Probability of Getting a Car by Not Switching Doors Increases by Switching Doors After Doors Have Been Opened to Reveal Goats Equation (4) can also be rewritten as follows. 1 1 scd Cddo (8) The first member of the right-hand term in Equation (8), cd, is the probability of getting a car by not switching. As was noted earlier, this probability is the same as the probability of initially picking a door hiding Copyright © 2011 SciRes. APM L. DEPUYDT 145 Ca car (i). The probability i C cannot change if one does nothing else. Equation (8) can therefore also be written as follows. 1 1 sid CC do (9) The second member of the right-hand term in Equa- tion (9), 1dd1o, is the factor by which the probability of getting a car by not switching doors is changed by actually switching. When there is only 1 car, then Equation (8) can be written in light of Equation (5) in §4.1 as follows. scg dgo C (10) The factor by which the probability of getting a car by not switching doors is changed by actually switching is then g go0o . If no doors are opened to reveal goats, then and the factor in question is 1d1d (or g g 1 when there is only 1 car), that is, 1. In this specific case, Equation (9) is equiv alent to the following Equation. s ii CC C 0 o (11) The probability does not change by switching doors and one hence does not increase o ne’s chances of gettin g the car by switching. If one or more doors are opened, then o and is smaller than . It follows that 1d1d 1dd1o is larger than 1 when any doors are opened. In this case, Equation (9) is equivalent to the following Equatio n. s i CC (12) In sum, one will always improve one’s chances of get- ting a car if one switches doors after doors have been opened to reveal goats. The increase in the probab ility of getting the car will vary. For example, it will be small if there are many, many doors and only a couple of doors hiding goats are opened. But there will always be an in- crease. What is more, any increase in the number of opened doors (o) produces an increase in the probability of get- ting the car by switching doors ( s C ). As o in 1dd1do 1o increases, decreases. Ac- cordingly, the fraction as a whole increases and so does s C. Any increase in the number of doors hiding cars (c) increases the chances of getting a car in genera l and any increase in the number of doors hiding goats (g) de- creases those chances in general. But any changes in the number of cars or doors do not affect the probability of getting a car by switching doors. Only the number of opened do ors (o) does. 4.3. Examples 4.3.1. Openin g Do o rs t o Reveal Goats Always Increases the Probability of Getting a Car by Switching Doors Equation (4) in §4.1 defines the probability that one will get a car by switching doors and makes it possible to compare this probability with the pro bability of getting a car by not switching. It is a fact that the probability of getting a car will always remain th e same when no doors are opened to reveal goats and will always in crease when one or more doors are opened. There is therefore never any point in switching doors to get a car when no doors are opened to reveal goats and always reason to switch when do ors are opened. The factor by which the probability in question in- creases by switching doors when doors are opened is 11ddo in general and g go in the special case in which there is only 1 car (§4.2). 4.3.2. The Special Status of the Monty Hall Problem The Monty Hall problem is just one instance of a much more general problem. The Monty Hall problem is spe- cial in that it presents what may be called the minimal scenario of the general problem. It exhibits the lowest numbers of doors, cars, and opened doors. The number of cars cannot be lowered because there is only one. The number of door s cannot be low ered fro m 3 to 2—and the number of goats therefore from 2 to 1—because it then becomes impossible to always open a door revealing a goat. By using Equation (4), it can be determined that the chances of getting the car by switching are as follows. 1131 2 133113 cd dd o Moreover, the probability of getting the car is in- creased through switching by a factor of 1 1 d do , in this case of 13 12 3311 . Because there is only 1 car, it is possible to use Equa- tion (5) in §4.1. The probability of getting the car by switching doors is then determined as follows. 22 32 13 g dg o Copyright © 2011 SciRes. APM 146 L. DEPUYDT In any variations on the Monty Hall problem, the numbers of do ors hiding cars and goats and opene d do or s can only be increased. Increasing the doors hiding cars increases the probability of initially picking a car and therefore also of getting a car by not switching. Increas- ing doors hiding goats decreases that same probability. But neither will increase the probability of getting a car by switching as op posed to not switching. Th at probabil- ity is increased only b y increasing the number of opened doors. 4.3.3. Chan ging the Number of Doors ( d) a nd Doors Opened to Reveal Goats (o) The most characteristic effect of the Monty Hall p roblem and its variations is the way in which opening doors to reveal goats increases the probability of getting a car by switching doors. What is more, the increase of the prob- ability of getting a car by switching doors grows as ever more and more door s are opened to reveal goats. But th is increase can only grow on condition that there are more doors hiding goats in the first place. For example, the number of doors hiding goats could be increased to 9 and the total number of doors therefore to 10. If only 1 door is opened to reveal a goat, the prob- ability of getting the car by switching doors ( s C) is, in light of Equation (4) in §4.1, as follows, given 1 car (c), 10 doors (), a n d 1 opened door (). do 11 11 cd dd o 101 9 0101180 or 11.25% Because there is only 1 car, it is possible to use Equa- tion (5) in §4.1. The probability of getting the car by switching doors is then determined as follows. 99 9 180 10 g dg o The probability of getting the car by not switching is 110 or 10%. The increase in the probability of getting the car by switching is only 1.25%. The difference be- tween switching and not switching would only become noticeable as the process of picking a door by switching doors is repeated over and over agai n. But as the opened doors (o) increase in number, so does the probability of getting the car by switching doors. If there are 2 doors, the probability is as follows. 99 9 270 10 g dg o or about 12.6% As 6 more doors hiding goats are opened one by one until only 1 unopened door hiding either a car or a goat remains, the probability of getting the car by switching doors gradually increases as follows: from 960 (15%) to 950 (18%) to 940 (22.5%) to 930 (30%) to 920 (45%) and finally to 910 (90%). When only 1 door is left unopened besides the initially picked door, the chances of getting the car are 9 times greater when switching doors than when not switching doors and one has a 90% chance if one switches. What is more, the probability of getting the car by switching doors no less than doubles when, of 2 doors remaining unopened be- sides the picked door, 1 is opened and only 1 door is left to which one can switch . If there ar e 1000 doors of which 1 hides a car an d 999 hide a goat and 998 doors are opened to reveal goats, the chance of getting the car by switching to the 1 door re- maining unopened is as follows. 999 999 1000 9999981000 g dg o One has a 99.9% chance of getting the car by switching doors. 4.3.4. Chan ging the Number of Cars (c) The examples provided have so far illustrated changing the number of doors hiding goats (g) and changing the number of opened doors (o). The main ai m of the exam- ples was to illustrate the effect of opening doors in terms of probability. Towards that end, the number of doors being opened to reveal goats had to be increased. But this number cannot be increased without increasing the number of doors hiding goats. That still leaves changing the number of cars to be illustrated. As was noted above (§4.1 end), when there is 1 car, there is one scenario, and when there is more than 1 car, there are two scenarios. As will be seen below, when there are two scenarios, the probability of getting a car by switching doors consists of a sum of two terms. When there is only one scenario, there is only one term. Let us assume that c = 6, g = 4, d = 10. The probability of initially picking a door hidin g car and th erefo re also of getting a car by not switching doors is 610 or 60%. The probability of getting a car by switching doors in- creases as follows when first 1, then 2, and finally 3 doors are opened to reveal a goat. opening 1 d oo r: 610 154 10 101180 or 67.5% opening 2 doo rs: 610 154 10 101270 or about 71.1% opening 3 d oors: 610 154 10 101360 or 90% Copyright © 2011 SciRes. APM L. DEPUYDT 147 i i 4.4. Probability Theory as a Mathematical Discipline That Combines the Digital or Non-quantitative with the Quantitative The design of §§4.4-7 is to clarify how Equation (4) (§4.1) has been obtained. This Equation makes it possi- ble not only to determine the probability of getting a car by switching doors under the precise conditions stipu- lated in the Monty Hall problem (see above) but also to extend and generalize the Monty Hall problem to any number of doors, cars, and doors that are opened to re- veal a goat. However, getting a car is only one possible outcome. The other outcome is not getting the car. Equa- tion (4) is therefore only one component of a larger mathematical problem. To ensure an adequate analysis and a true appreciation of the Monty Hall problem itself, it will be necessary to map the mathematical structure of the larger problem. The larger problem concerns all pos- sible outcomes . The total p robability of all outco mes is 1 or 100%. One or the other outcome of all possible out- comes must always be the case. But first, to ensure an even deeper appreciation of the Monty Hall problem on an even more general level, it will be useful to highlight a most peculiar general prop- erty of the type of mathematics needed to analyze the problem and its extensions, namely probability theory. In that regard, no one more than Boole has taken pains to emphasize—and in fact hardly anyone has in addition to him—that prob ability theory is a mathematical discipline straddling the digital or non-quantitative and the quanti- tative. The mathematical anatomy of the larger problem of which the Monty Hall is a specific case hence has two components, a digital or non-quantitative component and a quantitative component. The relation between the digi- tal component and the quantitative component is such that the digital component provides a skeleton that is fleshed out by the quantitative component. 4.5. The Digital or Non-quantitative Component in the Mathematical Mapping of the Monty Hall Problem and Its Extensions The larger problem of which the Monty Hall problem presents just one specific case revolves entirely around the four possible combination events in which either a first event or its supplement or negation can be followed by either a second event or its supplement or negation. In the specific case of the Monty Hall problem, the prob- ability of two of the four possible combination events is zero. Two classes, whether of things or of events, and their supplements produce exactly four combination classes (§2.6). Three events and their supplements would pro- duce exactly eight combination classes, and so on. This structure is digital or non-quantitative. One might rightly object t h at at l ea s t t h e n umb er o f th e co mb i nat i o n clas s es is quantitative. Strictly speaking, therefore, it is only the exploitation of the contrast between a class and its sup- plement and the combination of classes and their sup- plements into combination classes that is digital. Certain combinations of two classes and their supple- ments are invalid. Combining a class with itself produces just that class (§2.1) and is therefore not a valid combi- nation. Combining a class with its supplement is not pos- sible because, according to the fundamental axiom of thought, something cannot at the same time be and not be something (§2.8). In the case of the Monty Hall problem and any exten- sions thereof, the first event is initially () picking a door hiding a car (C), denoted here by i C. The first event can also be defined as failing to pick a door hiding a goat or a non-car. The supplement or negation of the first event is initially () failing to pick a door hiding a car (C), denoted here by i C. The supplement may also be described as picking a door hiding a non-car. The second event is picking a door hiding a car () by switching (s) doors and is denoted here by C s C. The supplement or negation of the second event is failing to pick a door hiding a car ( C) by switching (s) doors and is denoted here by s C is CC . In sum, there are exactly four digital com- bination classes (of events), the following. (1) (2) is CC (3) is CC (4) is CC The total probability of the occurrence of these four digital combination events is 1 or 100%. That means that one of these four combination events must always be the case. The four digital combination classes are also mutu- ally exclusive. Combining them would result in combi- nations of a class and its supplement. As has already been noted, according to the fundamental axiom of thought, such combinations are impossible. Something cannot at the same time be and not be something. Of the four digital combination classes, two have the desired outcome of picking a car by switching doors (C s ), namely (1) and (3). It is on these two combination classes that the study of the Monty Hall problem and any exten- sions thereof focuses (see §4.7 below). But the problem can only be mathematically appreciated to its full extent by considering all possible outcomes, whose combined probability is 1 or 100%. Copyright © 2011 SciRes. APM 148 L. DEPUYDT 4.6. The Quantitative Component in the Mathematical Mapping of the Monty Hall Problem and Its Extensions 4.6.1. Preamble: Boole’s “×” as an Expression of How the Digital and the Quantitative Complement One Another as Two Facets of a Single Phenomenon At the center of the mathematical structure of the larger problem of which the Monty Hall problem presents just one specific case stand four mutually exclusive digital combination classes of events. In Boole’s algebra, they are denoted by is CC , is CC, i CC s , and is CC CC . As a notation of digital combination classes, the sym- bol “×” represents the Boolean AND-operator, which is used for example when one searches for all that is both Paris and hotels on the Internet. However, in Boole’s algebra, the notations is , is , CCis CC, and is can also stand for the numerical probabilities of the occurrence of the four combination classes. The symbol “×” then functions as the multiplication sign of quantitative mathematics. Boole’s notation system therefore most felicitously repre- sents probability as a single phenomenon that exhibits digital or non-quantitative properties and quantitative properties inextricably linked to one another. CC The quantitative probability of a combination class of events involving two events is obtained by first calculat- ing the probability of the first event, then calculating the probability of the second event, and finally multiplying the two probabilities with one another. In calculating the probability of the second event, it needs to be taken in account that the second event may be dependent on the first event according to the defini- tion of dependence in classical probability theory. The dependence is such that the occurrence of the first event creates a new situation that influences the calculation of the probability of the second event. In other words, the second event takes place in a environment different from the one in which the first event takes place. And it is the first event that changed the environment. 4.6.2. The Probability of Digital Combination Event 1 1 cc o CC C dd ： is CC To establish the probability of is , the probability of the first event i initially (i) pickin g a door hiding a car (C), needs to be determined first. This probability is the ratio of all the cars to all the doors, which can be denoted by cd. If there is 1 car and there are 3 doors, the chance of picking the doo r hid i ng the car is 1 in 3. Next is establishing the probability of the second event s C 1c 1d , picking a door hiding a car (C) by switching doors (s). This probability is also a ratio of cars to doors. Since a car has already been picked in the first event, the num- ber of cars needs to be reduced by 1 to . And since one cannot pick again the door picked in the first event because one needs to switch, the number of doors needs to be reduced by 1 to . As was already noted above, the number of picked doors, 1, is not denoted algebrai- cally by , say , p because the form ula be i ng deve lo p ed here is only valid when 1 door is picked, that is, when 1p . Furthermore, since one cannot pick any doors being opened to reveal a goat, needs to be reduced fur- th er by t he num ber of op en ed doo r s (o) to . The 1d1do probability of s C will therefore be 1 1 c do is CC as a ratio of cars to doors. The combined probability of will be 1 1 cc ddo . In the case of the Monty Hall problem, the probability in question is as follows. 111 101 00 3311 313 This probability is zero. It is not possible to pick the door hiding the car by switching doors if one has already initially picked the car because there is only 1 car. 4.6.3. The Probability of Digital Combination Event is CC : 1 cgo ddo As regards the probability of is , the probability of the first event C has already been determined to be CC i cd(§4.6.2). Next is establishing the probability o f the seco nd ev ent s C, picking a door hiding a goat or non-car (C) by switching doors ( s). This probability is a ratio of goats to doors. Since no goat has been picked in the first event, the number of goats does not need to be reduced in this respect. However, sinc e one cannot pick any doors being opened to reveal a goat, the goats in those doors cannot be picked either and the number of the goats (g) needs to be reduced by the number of opened doors (o) to g o . Furthermore, again because one cannot pick any doors being opened to reveal a goat, the number of doors available for being picked () needs to be reduced as well by the number of opened doors (o), namely to 1d 1do . The probability of s C will therefore be g 1 o do as a ratio of goats to doors. The combined probability of is CC will be Copyright © 2011 SciRes. APM L. DEPUYDT 149 1 cgo ddo . In the case of the Monty Hall problem, the probability in question is as follo ws. 121 3311 3 111 1 1 1 33 In one third of all possible cases, one will initially pick a door hiding a car and then pick a door hiding a goat by switching doors. 4.6.4. The Probability of Digital Combination Event 1 gc dd o ： is CC To establish the probability o f is , the probability of the first event CC i C, initially (i) picking a door hiding a goat or non-car (C), needs to be determined first. This probability is a ratio of all the goats to all the doors, which can be denoted by g d. If there are 2 goats and there are 3 doors, the chance of picking a door hiding a goat is 2 in 3. Next is establishing the probability o f the seco nd ev ent s C 1d1d 1do , picking a door hiding a car (c) by switching doors (s). This probability is a ratio of cars to doors. Since no car has been picked in the first event, the number of cars (c) does not need to be reduced. But since one cannot pick again the door picked in the first event because one needs to switch, the number of doors needs to be reduced by 1 to . Furthermore, since one cannot pick any doors being opened to reveal a goat, needs to be reduced further by the number of opened doors (o) to . The probability of s C will therefore be 1 c do as a ratio of cars to doors. The combined probability of is CC will be 1 gc dd o . In the case of the Monty Hall prob lem, the probability in question is as follo ws. 21 3311 3 212 2 1 1 33 In two thirds of all possible cases, one will initially pick a door hiding a goat and then pick a door hiding the car by switching do o rs. 4.6.5. The Probability of Digital Combination Event 1 1 gg o dd o ： is CC As regards the probability of is , the probability of the first event Next is establishing the probability o f the seco nd even t s C, picking a door hiding a goat or non-car (C) by switching doors ( s). This probability is a ratio of goats to doors. Since a goat has been picked in the first event, the number of goats needs to be reduced by 1 to g 1 . Fur- thermore, since one cannot pick any doors opened to reveal a goat, the goats in those doors cannot be picked either and the number of the goats needs to be reduced further by the number of opened doors (o) to g o1 1d 1do . Furthermore, also because one cannot pick any doors being opened to reveal a goat, the number of doors available for being picked () needs to be reduced as well by the number of opened doors (o), namely to CC i C has already been determined to be g d(§4.6.4). . The probability of s C will be g 1 1 o do as a ratio of goats to doors. The combined probability of is CC will be g 1 1 go dd o . In the case of the Monty Hall prob lem, the probability in question is as follo ws. 2211 20200 3311 31 3 This probability is zero. There are 2 goats. It is not possible to pick a door hiding a goat by switching doors if one has already initially picked a goat because the other door hiding a goat is opened and the second goat is no longer available for picking. 4.6.6. The Combined Probability of All Digital Combinatio n E vents: 1 or 100% In a digital perspective, two classes and their supple- ments partition the un iverse or all that on e cou ld possibly think about into four combination classes (§2.6). The universe that is being considered in any partition often exhibits unspoken or explicit limits and is therefore not quite all that one could think about (§2.7). In a quantita- tive perspective, the numerical probabilities of all four combination events must add up to 1 or 100%. One or the other of the four combination classes must be the case and the four combination classes may take place with different degrees of probability. Accordingly, the following Equation ap plies. The notation is Boole’s. 1 111 11 1 cccgog c dd odd odd o gg o dd o (13) Because probability exhibits simultaneously a digital or non-quantitative facet and a quantitative facet, this Equation can be read in two different ways (§4.6.1). If “×” is interpreted as the Boolean AND-operator, the reading is digital and the four terms are combination Copyright © 2011 SciRes. APM 150 L. DEPUYDT classes of events, each characterized by the joint occur- rence (×) of two events. If “×” is interpreted as the mul- tiplication sign of quantitative mathematics, the four products (×) represent the numerical probabilities of the occurrence of each of the combination events. 4.7. The Probability of the Two Digital Combination Events Producing the Desired Outcome, Getting the Car, by Switching Doors, namely and is CCis CC The Monty Hall problem and any extensions thereof re- volve around a challenge with a desired outcome, namely getting a car. The question is whether the desired out- come can be achieved with greater or lesser probability by switching doors. The focus is therefore on determin- ing the probability of all the cases in which one gets the car by switching doors. It appears that only two of the four possible combination events discussed above (§4.6) have getting a car by switching doors as the outcome, namely is and CCis (§§4.6.2 and 4.6.4). The combined probab ility of these two combination ev ents is as follows (§§4.6.2 and 4.6.4). CC 1cc ddo 11 g c dd o 10c (14) If there is only 1 car, then and therefore 10 1do c and 10 1 cc ddo . The probability of getting a car by switching is then as follows. 1 gc dd o The two products making up expression (14) have the same denominator. Expression (14) can therefore be re- written as follows. 1 1 cc gc dd o And, in light of the common factor c, also as follows. 1 1 cc g dd o And furthe r as fol lo w s. 1 1 cc g dd o cgd And since , the following expression is also equivalent. 1 1 cd dd o This is the probability that switching doors will pay off . Extension and Generalization of the of .1. “Doubling” the Monty Hall Problem ery many are the ways in which the Monty Hall prob- e than one door is pi .2. The Increase in Digital Complexity ecause probability is a phenomenon exhibiting both e case in which 2 doors are picked initially, as by getting a car, as stated in Equation (4) in §4.1. 5Monty Hall Problem to Any Number Picked Doors (p) 5 V lem can be extended. One might for example consider cases in which one can pick three or more types of things hiding behind doors, cars and goats and other types of things. Exploring more of these extensions remains de- sirable. But what can be done within the confines of the present paper is limited. Moreover, a principal aim of the present paper is highlighting the digital component of the analysis of the problem and its extensions in its relation to digitality as a fundamental component of human cog- nition. A discussion of countless extensions of the Monty Hall problem would probably not shed much additional light on the fundamental assumption made here as a working hypothesis, namely that rational thought and language is profoundly digital. In what follows, the analysis of extensions will be limited to cases in which more than one door can be picked. One specific case in which mor cked involves “doubling” the Monty Hall problem in all its characteristics. Accordingly, there would be 2 cars instead of 1, 4 goats instead of 2, 6 doors instead of 3, and 2 doors would be opened to reveal goats instead of 1. Furthermore, the question would now be: If one picks 2 doors that remained closed and 2 doors are opened to reveal goats, should one switch to the 2 remaining un- opened doors to improve one’s chances of getting 2 cars and, if so, by how much would one improve one’s chances? 5 B digital or non-quantitative and quantitative properties (§§3.1 and 4.6.1), making the number of picked doors into a variable will add complexity to the analysis of the extended Monty Hall problem. As regards the digital complexity, the original Monty Hall problem and the extensions to any number of doors, cars, and opened doors discussed above involve only digital combination events. The additional extension to any number of picked doors involves digital combinations of digital combina- tion events. Consider th in the “doubling” of the Monty Hall problem (§5.1). In the original Monty Hall problem and its extension to Copyright © 2011 SciRes. APM L. DEPUYDT 151 lly picked and 2 do ing, tw any number of doors, cars, and opened doors discussed above, the initial pick of a door and the pick of a door by switching doors are both single events. By contrast, when 2 doors are initia ors are picked after switching, both the initial pick an d the pick by switching are composite. Both consist them- selves of combination events. There are 2 initial picks that follow one another in sequence and 2 picks by switching that also follow one another in sequence. In both the 2 initial picks and the 2 p icks by switch o classes of events and their supplements generate four digital combination classes (§2.6). As regards the 2 ini- tial picks, the two classes of events are picking a car (C) in the first ( f ) initial (i) pick ( f i C) and picking a r (C) in the second (ca s ) nitial ( iick (i) p s i C). The two supplement classes are failing to pick a c(ar C) in the first (f) initial (i) pick ( f i C), or picking a goatnd fail- ing to pick a car (, a C) ine second ( th s ) initial (i) pick ( s i C), or picking a goat. Accordingly, the four digital cobination classes characterizing just the 2 initial picks are as follows: m f isi CC, f isi CC, f isi CC, and f isi CC. gardAs res the 2 picks by switching doors after doors have been opened to reveal goats, the two classes of events are picking a car ( C) in the first ( f ) pick by switching ( s ) ( f s C) and picing a car (C) inthe second (k s ) pick byswiing ( tch s ) ( s s C). The splement of the first event is failing to pick ar (up a cC) in the first ( f ) pick by switching ( s ) ( f s C), or pickg a goat. The su- plement of the secot is failing to pick a car ( in p ennd evC) in the second ( s ) pick by switching ( s ) ( s s C), or pi- ing a goat. Accordingly, the four diital mbination classes characterizing the 2 picks by switching doors are as follows: ck cog f sss CC, f sss CC, f sss CC, and f sss CC. .3. The 16 Digital Combinations of the ach of the four combination events of the in itial pick s is ars p = doors picked in the initial picks e 5“Doubled” Monty Hall Problem E combined with each of the four combination classes of the picks by switching. The result is 16 combinations of combination events. The 16 combinations are listed be- low along with the numerical probability of their occur- rence. c = c g = goats o = doors opened to reveal a goat after th intial picks (1) : 12 fi sifs ss CC CC cc c dd dpod 3 0 11 c po (2) : 12 0 11 fi sifs ss CC CC cg cc dd dpodpo (3) : 12 0 11 fi sifs ss CC CC gc cc dd dpodpo (4) : 11 11 4321 6 6521 15 fi sifs ss CC CC ggcc dd dpodpo (5) : 12 0 11 fi sifs ss CC CC cc cgo dd dpodpo (6) : 11 11 24112 6521 15 fi sifs ss CC CC cg cgo dd dpodpo (7) : 11 11 42112 6521 15 fi sifs ss CC CC gc cgo dd dpodpo (8) : 12 0 11 fi sifs ss CC CC ggc go dd dpodpo (9) : 12 0 11 fi sifs ss CC CC ccgo c dd dpodpo Copyright © 2011 SciRes. APM 152 L. DEPUYDT (10) : 1 24112 6521 15 fi sifs ss CC CC cg go dd dpo 11 1 c dpo (11) : 42112 6521 15 fi sifs ss CC CC gc go dd dpo 11 11 c dpo (12) : 12 1 fi sifs ss CC CC gg go dd dpod 0 1 c po (13) : 1 2121 1 6121 15 fi sifs ss CC CC cc go dd dpo 11 1 go dpo (14) : 1 fi sifs ss CC CC cg go dd dpod 12 0 1 go po (15) : fi sifs ss CC CC g cgo dd dpod 12 0 11 go po (16) : 12 1 fi sifs ss CC CC gg go dd dpod 3 0 1 go po probability of 10 combinations of combination events is zero. In (1), (2), (3), (5), and (9), cars are picked in 3 or 4 of the 4 picks, but there are only 2 cars. In (8), (12), (14), (15), and (16), goats are picked i3 or 4 of the 4 picks, but it is not possible to pick more than 2 in total. There are two scenarios. First, in (8), (12), and (1 rd on the first two, and th e fourth on the first ree. For example, in (4), a door hiding a go at is in itially ent. The chance of picking a goat is The n 6), 2 goats are initially picked. When 2 doors are then opened to reveal goats, there are no goats left to pick. Yet, 1 goat is picked according to (8), (12), and (16). This is not possible. Second, in (14) and (15), 1 goat is initially picked. When 2 doors are then opened to reveal goats, only 1 goat is available for picking. Yet, 2 goats are picked according to (14) and (15). This is also not possible. 5.4. Dependence Each combination in §5.3 consists of a sequence of four events, with the second event being dependent on the first, the thi th picked in the first ev cd. In the second event, a door hiding a goat is again picked. But the number of the goats and the doors has been reduced by 1 in the fi rst event. The chance of again picking a goat in the second event is therefore 11cd . In the third event, a car is pick ed. But the number of doors has been reduced by the 2 doors that are ed in the first and second events (p), though not the number of cars as no car has been picked yet. In addition, the number of doors is reduced by 2 as 2 doors are opened to reveal a goat (o). The chance of picking a car pick is therefore 2cdpo . Finally, in the fourth event, a car is again picked. But, the number of cars has been reduced by 1. So has again the number of doors. The chance of picking a car is therefore 31cdpo . When goats are picked by swit- ching doors, that is, in the third and fourth component events, the n be reduced not only by goats that are picked but also by goats revealed by open- ing doors (o). y of a Successful Outcome by Switching Doors in the “Doubled” Monty Hall Problem Is one more likely or less likely of getting 2 cars by switching doo rs umber of goats can 5.5. The Probabilit than by not switchi n g do ors after 2 doors goat abili 2 cars by not switching and by switching need to be compared. have been opened to reveal goats if there are 2 cars, 4 s, and 6 doors? To answer this question, the prob- ties of getting the The probabilit y of getting the 2 cars by not switching is the same as the probability of initially getting 2 cars by picking 2 do ors. The pro bability o f pickin g 1 of th e cars at first pick is cd or 26 or 13. If one was successful in picking a car at first pick, then not only the number of the doors but also the number of the cars is reduced by 1. The probability of picking a car again at second pick is therefore 11cd or 21 61 or 15. The probability of picking both cars in the first 2 picks, then, is therefore 13 15 or 115 r about 6.6%. As for the probability of getting the 2 cars by switching o Copyright © 2011 SciRes. APM L. DEPUYDT 153 ars are pickednly 1 of the 1ita ce of th from the 2 doors initially picked to the 2 only doors that remain closed after 2 doors have been opened to reveal 2 goats, 2 c in o6 digl com- binations listed in §5.3, namely (4). The probability of the occurrenis coation is mbin615 or 25 or 4 s g it after switching doors. The robability of gettine car is 0%. Remarkably, one can only get the 2 cars by swit- ching doors if one had originally picked 2 doors hiding goats or no cars at all. 5.6. Comparison of the Original and the “Doubled” Monty Hall Problem In the original Monty Hall problem, one is more likely to get the car than not gettin g thp23. By contrast, in et the 2 cars doors. The robability of g is “doubling” the problem, one is less likely to g than not getting them after switching petting them615. On the other hand, in the original Monty Hall problem, one doubles one’s chances to get the car from 13 to 23 by switching doors. By contrast, in “doubling”e problem, one’s chances of getting both cars increase sixfold from th115 to 615. 5.7. Compressing Probability as an Effect of Opening Doors to Rev Goa As always, opening doors to reveal goats has the e of press ealts ffect coming or condensing a greater robability to wer doors (§3.3). There is a probf only p ability ofe115 5.5). How of getting both cars by initially picking 2 doors (§ ever, there is a probability of 615 or 40% th at the cars are both hiding behind 2 of the other 4 doors that rs 1 nd 4 as minator ll 16 digital combination events listed in §5.3 exhibit mber of pick 2 one has not picked. This number is obtained as follows. There are 15 different ways in which the 2 cars can be hiding behind the 6 doors: behind doors 1 and 2, doo and 3, doors 1 and 4, doors 1 and 5, doors 1 and 6, doors 2 and 6, doors 2 and 3, doors 2 a, doors 2 and 5, doors 2 and 6, d oors 3 and 4, doors 3 and 5, door s 3 and 6, doors 4 and 5, doors 4 and 6, and doors 5 and 6. Let us sume that the doors picked initially are 1 and 2. In 6 out of a total of 15 ways, or 40%, the 2 cars are hiding behind the 4 other doors, namely doors 3 to 6. They are the last 6 locations in the list just provided. In 9 of 15 ways, at least 1 car is hiding behind either door 1 or door 2 or both. 5.8. Denominator and Numerator of the Probabilities Involved in the “Doubled” Monty Hall Problem 5.8.1. Den o A the same denominator. Furthermore, as the nu s increases, the denominator will morph into 123d ddd 1 3 dp odp o po 2dp o d as fol- lows. This progression is obviously factorial, as expressed by “!”. For any number of doors (d), opened doors (o), and picked doors (p), the numerator will therefore be ! !dpo d ! dp dpo p ! The details exceed the scope of the present paper. Suf- fice it to note that 0!, which is obtained when 0o pdp , is not the same as zero or nothing but rather signifies the absence of any additional factor. 5.8.2. Numera tor As of what is sought. The que g doors. But what if one took satisfaction with getting s one improve one’s chances in that s regards the numerator of the 16 combination classes listed in §5.3, an evaluation is necessary in term stion was asked before how much one improves one’s chances of getting both doors by switch- in just 1 car? How doe case by switching both doors? One’s initial chances of getting at least 1 car by picking 2 doors is the combined probability of the three combined events of picking first a car and then again a car, of picking first a car and then a goat, and of picking first a goat and then a car, or 1 111 ccc gg c dd dddd , in this case 212442 9 65656 515 . As regards switching, the outcome of five of the 16 combination classes listed above includes getting at least 1 car. They are combina- tion classes (4), (6), (7), (10), r combined probability is and (11). Thei 14 15. In sum, one does not double one’s chances of getting at leas the probability of not getting one has been reduced to as t 1 car by switching doors. But little as 115. 6. Conclusions Walking in the footsteps of Boole has made it possible, I believe, to construct a map of the mathematical structure of the M Hontyall problem in its context and of certain s thereof, ev en if the analysis rmutations remains desirable. Two mponents, one digital-mathematical or non-quantitative extensions or generalization of even higher pe co and the other quantitative, complement one another to make up this map. But the focus of the present paper is not only on this mathematical structure but also on its Copyright © 2011 SciRes. APM L. DEPUYDT Copyright © 2011 SciRes. APM 154 ng Professor f Finance at Tshinghua University’s School of Ec r reading and comme n the present paper. In explaining the Monty Hall prob- a lecture entitled “How the Bio- lo to, in the fr ] J. Gill, “Bayesian Methods,” International Encycloped ndica, Vol. 65, No. 1, 2010, pp. 57-71. relation to the presumed digital nature of cognition as expressed in rational thought and language. 7. Acknowledgements I thank Dr. Michael R. Powers, Professor of Risk Man- agement and Insurance at Temple University’s Fox School of Business and Distinguished Visiti o o- nnomics and Management, foting o lem to students, Professor Powers prefers to have re- course to the method known as conditional probability. But he also believes that different approaches may sup- plement one another. I am grateful to Prof. Dr. Andreas Manz, Head of Re- search at the Korean Institute of Science and Technology (KIST) at the University of Saarbrücken, Germany, for inviting me to participate in a workshop held at KIST on June 30-July 2, 2010 (see www.humandocument.org ). At this workshop, I read gical Brain Reasons: The Four Digital Operations Un- derlying All Rational Language and Thought.” The lec- ture concerned the digital analysis of rational thought and language. The ideas presented therein have inspired the present paper on the Monty Hall problem. I was also grateful for the opportunity to be able to present some ideas on the digitalization of the analysis of rational tho ught and language on April 28, 2011 in a lec- ture entitled “The Inevitable Digitization of Language Analysis” and read in the Department of Near & Middle Eastern Civilizations of the University of Toron amework of an Information and Discussion Session on the topic “Does It All Add Up? Quantitative Reasoning (QR) in the Humanities.” Finally, I thank two anonymous reviewers of an earlier version of this paper for their critical and penetrating comments. These comments have necessitated a com- plete overhaul of the paper. 8. References [1] J. 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However, because of the perfect duality inherent in the algebra, the postulates 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, Pis- cataway, New Jer [9] J. Venn, “Symbolic Logic,” 2nd Edition, Macmillan and Company, London and New York, 1894, pp. 245-255. [10] E. Schröder, “Vorlesungen über Die A (Exakte Logik),” J. C. Hinrichs, Leipzig, Vol. 1, 1890, p. 319. [11] L. Depuydt, “Zur Unausweichlichen Digitalisierung der Sprachbetrachtung: ‘Allein,’ ‘anderer,’ ‘auch,’ ‘einziger,’ ‘(sein tisch-Koptischen und im Allgemeinen” (“On the Unavoid- able 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 Monas- teriensia as part of the acts of the Workshop “Lexical Fields, Semantics and Lexicography” held 5-7 November, 2010 at the University of Münster, Germany. [12] L. Depuydt, “The Other Mathematics: Language and Logic in Egyptian and in General,” Piscataway, New Jer- sey, 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 |