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Advances in Pure Mathematics, 2012, 2, 243-273 http://dx.doi.org/10.4236/apm.2012.24034 Published Online July 2012 (http://www.SciRP.org/journal/apm) 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 Leo Depuydt1,2, Richard D. Gill3 1Department of Egyptology and Ancient Western Asian Studies, Brown University, Providence, USA 2Department of the Classics, Harvard University, Cambridge, USA 3Mathematisch Instituut, Universiteit Leiden, Leiden, The Netherlands Email: leo_depuydt@brown.edu, gill@math.leidenuniv.nl Received January 13, 2012; revised March 13, 2012; accepted March 25, 2012 ABSTRACT In Advances in Pure Mathematics (www.scirp.org/journal/apm), Vol. 1, No. 4 (July 2011), pp. 136-154, the mathe- matical structure of the much discussed problem of probability known as the Monty Hall problem was mapped in detail. It is styled here as Monty Hall 1.0. The proposed analysis was then generalized to related cases involving any number of doors (d), cars (c), and opened doors (o) (Monty Hall 2.0) and 1 specific case involving more than 1 picked door (p) (Monty Hall 3.0). In cognitive terms, this analysis was interpreted in function of the presumed digital nature of rational thought and language. In the present paper, Monty Hall 1.0 and 2.0 are briefly reviewed (§§2-3). Additional generaliza- tions of the problem are then presented in §§4-7. They concern expansions of the problem to the following items: (1) to any number of picked doors, with p denoting the number of doors initially picked and q the number of doors picked when switching doors after doors have been opened to reveal goats (Monty Hall 3.0; see §4); (3) to the precise condi- tions under which one’s chances increase or decrease in instances of Monty Hall 3.0 (Monty Hall 3.2; see §6); and (4) to any number of switches of doors (s) (Monty Hall 4.0; see §7). The afore-mentioned article in APM, Vol. 1, No. 4 may serve as a useful introduction to the analysis of the higher variations of the Monty Hall problem offered in the pre- sent article. The body of the article is by Leo Depuydt. An appendix by Richard D. Gill (see §8) provides additional context by building a bridge to modern probability theory in its conventional notation and by pointing to the benefits of certain interesting and relevant tools of computation now available on the Internet. The cognitive component of the ear- lier investigation is extended in §9 by reflections on the foundations of mathematics. It will be proposed, in the foot- steps of George Boole, that the phenomenon of mathematics needs to be defined in empirical terms as something that happens to the brain or something that the brain does. It is generally assumed that mathematics is a property of nature or reality or whatever one may call it. There is not the slightest intention in this paper to falsify this assumption because it cannot be falsified, just as it cannot be empirically or positively proven. But there is no way that this assumption can be a factual observation. It can be no more than an altogether reasonable, yet fully secondary, inference derived mainly from the fact that mathematics appears to work, even if some may deem the fact of this match to constitute proof. On the deepest empirical level, mathematics can only be directly observed and therefore directly analyzed as an activity of the brain. The study of mathematics therefore becomes an essential part of the study of cognition and human intelli- gence. The reflections on mathematics as a phenomenon offered in the present article will serve as a prelude to planned articles on how to redefine the foundations of probability as one type of mathematics in cognitive fashion and on how exactly Boole’s theory of probability subsumes, supersedes, and completes classical probability theory. §§2-7 combined, on the one hand, and §9, on the other hand, are both self-sufficient units and can be read independently from one an- other. The ultimate design of the larger project of which this paper is part remains the increase of digitalization of the analysis of rational thought and language, that is, of (rational, not emotional) human intelligence. To reach out to other disciplines, an effort is made to describe the mathematics more explicitly than is usual. Keywords: Artificial Intelligence; Binary Structure; Boolean Algebra; Boolean Operators; Boole’s Algebra; Brain Science; Cognition; Cognitive Science; Definition of Mathematics; Definition of Probability Theory; Digital Mathematics; Electrical Engineering; Foundations of Mathematics; Human Intelligence; Linguistics; Logic; Monty Hall Problem; Neuroscience; Non-Quantitative and Quantitative Mathematics; Probability Theory; Rational Thought and Language C opyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL Copyright © 2012 SciRes. APM 244 1. Introduction In Advances in Pure Mathematics, 2011, Vol. 1, No. 4, pp. 136-154, the mathematical structure of the well- known problem of probability known as the Monty Hall problem (see §2 below) was mapped in detail [1-4]. This mathematical structure includes two components that complement one another seamlessly. One component is digital or non-quantitative. The other is quantitative. The focus of that earlier paper was mainly on the neglected digital component. The digital component was analyzed in the spirit and the algebra of George Boole’s Investiga- tion of the Laws of Thought (1854), the Magna Charta of the digital age. Much of what has been said in the earlier paper is presupposed in what follows. In said article, the analysis of the Monty Hall problem was extended in two directions. First, on the cognitive side, the digital analysis was interpreted as an organic reflection of the presumed digital nature of human cogni- tion as expressed by rational thought and language and as evidenced empirically by facts of language. Probing the nature of rational thought and language was in a sense the ulterior motive of analyzing the Monty Hall problem. Second, on the mathematical side, the Monty Hall prob- lem was generalized to related cases in accordance with the axioms of probability theory (Monty Hall 2.0). The aim was to demonstrate the reliability and productivity of the proposed digital approach. This first generalization is briefly reviewed in §3 below. The analysis of the Monty Hall problem is extended again, both mathematically and cognitively, in the pre- sent paper. First, in mathematical terms, the validity of the proposed digital approach is bolstered by additional generalizations of the Monty Hall problem in §4, §5, §6, and §7 (Monty Hall 3.0 and 4.0). This process could presumably be carried on ad infinitum, at some point entering the domain of calculus. Second, in cognitive terms, an attempt is made to ren- der the presumed deep organic link between the digital component of probability theory and the digital nature of rational thought and language more probable by defining what mathematics is (see §9 below). In terms of the search for the deepest foundations of mathematics, it is proposed that mathematics is best defined first and fore- most as something that the brain does as it engages real- ity outside itself through the senses. §§2-7 combined, on the one hand, and §9, on the other hand, are self-sufficient and can be read independently from one another. In other words, it is not necessary to read §§2-7 in order to read §9. An appendix by Richard D. Gill (see §8) provides ad- ditional context by building a bridge to modern probabil- ity theory in its conventional notation and by pointing to the benefits of certain interesting and relevant tools of computation now available on the Internet. It is hoped that the reflections presented in §9 on the nature and definition of mathematics will serve as a prelude to forthcoming papers on the foundations of probability theory as one type of mathematics entitled “How Boole’s Theory of Probability Subsumes, Super- sedes, and Completes Classical Probability Theory: A Digital, Quantitative, and Cognitive Analysis,” in which an attempt will be made to describe how exactly Boole’s theory of probability, which has been almost entirely neglected for one and a half centuries, makes the classi- cal theory of probability complete. It is imperative that a mathematical theory consider all possible cases. Classical probability theory does not. H.H. Goldstine writes about Boole that “our debt to this simple, quiet man... is extraordinarily great and probably not adequately repaid” [5]. Goldstine is refer- ring to the enormous significance of Boole’s digital mathematics in modern computer science. It is suggested in §9 that the extent of the debt may far exceed computer science and reach deeply into the analysis of rational thought and language or human intelligence. 2. Monty Hall 1.0: The Original Monty Hall Problem, Featuring 1 Car (c), 3 Doors (d), 1 Opened Door (o), 1 Door Initially Picked (p), and 1 Door Picked by Switching (q) Behind 3 closed doors, 2 goats and 1 car are hiding. One picks 1 door with the aim of getting the 1 car. The 1 door that one picks remains closed, however. Next, someone who knows what is hiding behind all the doors opens 1 of the 2 doors that were not picked, more specifically 1 door hiding a goat. 2 doors remain closed and available for picking, including the one initially picked. The Monty Hall problem involves the following question: Should one switch from the unopened door that one initially picked to the other door that remains unopened to im- prove one’s chances of getting the car? The answer is: One should, because one doubles one’s chances of get- ting the car—namely from 1 in 3 to 2 in 3—by switching doors once 1 door has been opened to reveal 1 goat. 3. Monty Hall 2.0: Generalization to Any Number of Doors (d), Cars (c), and Opened Doors (o) The present generalization is treated in detail in the arti- cle mentioned in §1 above. What follows is a brief sum- mary of this treatment. The Monty Hall problem involves 1 car (c), 2 goats (g), 3 doors (d), 1 opened door (o), and 1 picked door (p). There are 5 variables. But in extending and generalizing the Monty Hall problem, only 4 variables need to be considered. That is because, of the 3 variables c, g, and d, L. DEPUYDT, R. D. GILL 245 each can be derived from the two others. From the fact that cgd, it follows that cdg and g dc . Only 2 of the variables c, g, and d therefore need to be considered. In what follows, c (cars) and d (doors) are chosen. As a general rule, in Monty Hall 2.0, one always im- proves one’s chances of getting a car by switching doors when doors are opened to reveal goats. This will no longer be the case from Monty Hall 3.0 onward (see §4.12 and §6). The question remains: By how much? If the Monty Hall problem is generalized to any number of cars (c), doors (d), and opened doors (o), and only 1 door is picked, the chance of getting the car (C) by switching (s) doors (Cs) is 1 1 cd dd o , (1) and the factor by which one improves one’s chances of getting the car by switching is 1 1 d do . (2) The number 1 in these expressions represents the number of picked doors (p), which is fixed at 1. For example, let there be 123,456,789 (or more than 123 million) doors (d), of which 12,345,678 (or more than 12.3 million) hide cars (c). Also assume that 1,234,567 (or more than 1.23 million) doors are opened (o) to reveal goats. The chances of getting a car (C) by switching (s) doors (Cs) is, according to expression (1), 12,345,678123,456,7 123,456,789123, 456,7891 about0.101 or 10.0%. 89 1 1, 234,567 The factor by which one increases one’s chances of getting a car by switching doors is, according to expres- sion (2), 123,456,7891 123,456,78911, 234,567 about 1.010. If this factor were 1, one would not increase one’s chances because multiplying any number by 1 does not increase that number. But because the factor is about 1.010, one increases one’s chances by about 0.01 or about 1%. One’s chances of getting a car when initially picking 1 door is the fraction of which the number of cars (c) is the numerator and the number of doors (d) the denominator, namely c/d, which in this case is 12,345,678 123,456,789 1 = about 0.0999999927 or just about 10%. Increasing one’s chances from about 10% to about 10.1% indeed involves an increase of 1%, since 1% of 10 is about 0.1. Since there are 123,456,789 doors (d) and 12,345,678 cars (c), there are 111,111,111 goats (g). According to the rules of the extended Monty Hall problem, up to g doors can be opened to reveal goats, that is, 111,111,110 doors can be opened (o). If one opens the maximum number of doors that one is allowed to open, then according to expression (2) one increases one’s chances of getting a car by switching by a factor of 123,456,789112,345,678. 123,456,7891111,111,110 Since a factor of 1 corresponds to a 0% increase, a factor of 2 to a 100% increase, a factor of 3 to a 200% in- crease, and so on, a factor of 12,345,678 corresponds to an increase of 1,234,567,700%. In other words, one im- proves one’s chances of getting a car by more than 1.23 billion percent by switching. As regards the basic treatment of the Monty Hall problem in the afore-mentioned article, an additional note on notation is in order. Boole never ceased to im- press upon his readers that probability is a field of mathematics that straddles the digital-mathematical and the quantitative-mathematical. The digital-mathematical and the quantitative-mathematical coexist in the single phenomenon of probability. To use a metaphor, it is a bit like Christianity’s Trinity, three divine entities coexisting as one, although in this case not a trinity but a Duality is concerned. In probability as a field of mathematics, the digital-mathematical and the quantitative-mathematical are two facets of what is ultimately a single thing. Natu- rally, the human brain cannot quite think about the two facets at the very same time. But that is just a limitation of our mental capacities. 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]: :1 is cgo CCdd o . Both expressions describe the probability of initially picking a car and then picking a goat or non-car by switching. The expression to the left of the colon is digital- mathematical. In this expression, the quantitative aspect is irrelevant. Accordingly, the symbols Ci and s C are not quantitative. Likewise, if one divides the universe in strictly digital terms into four digital combination classes Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 246 involving the two classes “black” (b) and “cat” (c), then the universe (1) equals bcbc bcbc, that is, black cats, non-black cats, black things that are not cats, and things that are neither black nor cats. The sets bc “black cats” and bc “non-black cats” will in all probability differ in quantity, assuming that it is possible to count all black and non-black cats. However, the difference in quantity is irrelevant in the digital-mathematical expres- sion of the universe. The expression to the right of the colon is quantita- tive-mathematical. Indeed, the symbols c, g, d, and o are quantitative. They stand for numbers of cars, goats, doors, and opened doors. It follows that the symbol × admits of both a digital and a quantitative interpretation. The two interpretations may denoted by d and . Accord- ingly, the following equation applies: q quantitative . digital 1 id s q CC cgo dd o Multiplication is commutative. That means that a × b = b × a. However, it may be tempting to assume that d is not commutative. It is a fact that the event s C, not getting a car by switching doors, follows the event Ci, initially getting a car, in time. And yet, in contemplating the combination of Ci and s C, nothing prevents one from contemplating s C first. The order in which one contemplates the two does not matter mathematically, even if it may come more naturally to think first of what comes first in time. Likewise, on the quantitative- mathematical level, the following equation applies: 11 qq goc d od cgo dd o . 4. Monty Hall 3.0: Additional Generalization to Any Number of Doors Picked Initially (p) or of Doors Picked by Switching (q) 4.1. The Special Case of Getting at Least 1 Car When Switching Doors In Monty Hall 1.0 and 2.0, just 1 door is picked both be- fore and after switching. The most natural expansion of 1.0 and 2.0 would seem to be the generalization in which any number of doors are picked both before and after switching. The number of doors picked will be denoted by p; the number of doors picked by switching, by q. The present generalization is styled here as Monty Hall 3.0. One can imagine many desired outcomes of picking 1 or more doors. For example, the desired outcome might be to get cars with every door pick both before and after switching. Or the desired outcome might be to obtain 1 car in the 1st and the 3rd of 3 initial picks as compared to picking 1 car in the 1st of 2 picks by switching. And so on. Treating all desired outcomes comprehensively ex- ceeds the scope of the present paper. In such a compre- hensive treatment, it is necessary to take one’s departure from the equation representing the total probability of all possible outcomes, whose individual probabilities add up to 1 or 100%. It is hoped that it will be possible to pre- sent a survey of the respective probabilities of all possi- ble outcomes in a future paper. Presently, just 1 desired outcome will be selected. The aim is to select an outcome that concords with the spirit of the original Monty Hall problem. In the original prob- lem, the person picking a door wants a car. Accordingly, when more than 1 door is picked, the desired outcome that most closely reflects the spirit of the original Monty Hall problem is getting at least 1 car. It would be awk- ward to deny the person any car at all if more than 1 car is picked. The probability P that one will get at least 1 car by switching doors is a fraction whose numerator is N and whose denominator is D. N and D are defined below. Most the rest of §4 is devoted to a description of how the equation below is obtained. A more explicit version of the numerator appears in §4.17 below. The precise relation between the following expression and the common probabilistic conceptualization known as hypergeometric distribution will be described in a fu- ture paper. N PD !! ! !! 1! cp cp c Nq cp cpqcp q ! 1! go pp goppq q 1! 12 2! qqc p cp q ! 2! go pp goppq q 12 ! 123 3! qqqc p cp q ! 3! go pp goppq q 123 2 1232 1 qqqq qq q qq ! 1! cp cpq q Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 247 ! 1!goppqqq go pp !! 1! cg p 1! pcpgp 1! 1! 1! 1! gop p cp cp q g opp qq 1! 11! cp q qcp 1! 11! gop p goppq q 1! 12! cp q 1 12 qq cp 1! 12!q q gop p go p p 1! 13! cp q 12 123 qq q cp 1! 13!q q gop p gop p 12 123( 2) qqqq q qq 3 2 1 q q 1! 11!qq cp cp 1! 11! gop p goppq q q 1 12 2! pp cp gp !! 2! cg p 2! 2! cp cp q 2! 2! gop p g opp qq 2! 21! cp q qcp 2! 21!q q gop p go p p 2! 22! cp q 1 12 qq cp 2! 22! gop p goppq q 2! 12 123 23! cp qq q cp q 2! 23! gop p goppq q 123 2 1232 1 qqqq qq q qq 2! 21! cp cp qq 2! 21! gop p goppq q q 12 ! 123 3! pp pc cp ! 3! g gpp 3! 3! 3! 3! gopp cp cpq g opp qq 3! 31! cp qcp q 3! 31! gop p goppq q 3! 1 12 32! cp qq cpq 3! 32! gopp goppq q 3! 12 123 33! cp qq q cpq 3! 33! go p p goppqq 1232 1232 1 qqqq qq q qq 3! 31! cp cp qq 2! 21! gopp goppqqq Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 248 12 123 1 pppp p pp 2 1p p !! ! cg pp !cp pgp ! ! cpp cppq ! ! ppgop g opp p qq ! 1! cpp p q qcp ! ! goppp gopppq q 1 ! 1 12 2! cpp qq cpp q ! 2! p q q gop p gop pp ! 3! cpp p q 12 123 qq q cp ! 3! p q q gop p gop pp 12 123 2 qqqq q qq 3 2 1 q q ! 1! p qq () cp cp p ! 1!q goppp gopppqq ! !! dpo dpoq In the following maximally compacted version of the numerator, the coefficient terms have been reduced from 5 to 4 and much of the transparency has been lost. !d Ddp ! ! c Ncp !!! !1!1! cpcp go q cpq cpqgo !!cp go 2! 2!cpq go 12 2 123 1 qq q q !! 1! 1! cp go cp goq !! 1! 1! cg pcp g 1!1! 1! 1!2!2! cpcp go q cpq cpgo q 11!1! 1 23!3! qqc po cpq o g g 12 2 123 1 qq q q 1! 1! !! cp go cp goq 1!! 1 22!2! pp cg cp g 2!2! 2! 2!3! 3! cpcp go q cpqcpq go 12!2! 1 24!4! qqc po cpq go g 12 2 123 1 qq q q 2! 2! 1! 1! cp go cp goq 12 1 ! 123 ! pp pg pgp ! !! !1!1! p cc q cq cqgop go 1! ! 1 22!2! qqg p c cq gop o 12 ! ! 123 3! 3! qqqg op c cq gop 12 2 123 1 qq q q ! ! 1! 1! gop c cgopq 4.2. Point of Departure: An Example It will be convenient to begin the description of how the Equation in §4.1 is obtained with a specific example. Once it is seen how the probability of getting at least 1 Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 249 car by switching doors after doors hiding goats have been opened is obtained in 1 case, the result can be general- ized to all cases. In the example that will be used here, ors Hiding Cars or Goats It bability that a number of dif- fewith its own de- grand, all possible sc of doors that hide ei picks of doors (p = rs (q (o = 2). r); 4) cccgg; 5) ccgcc; 6) ccgcg; 7) ccggc; 8) cgcgg; 13) cggcc; 18) nu the example at hand, the chance of picking a car is c/d, , that is, 7/ her words, the probability of a later pick of either a goat or a car is dependent on what hap- probearlier events the numbers of the variables are as follows: cars (c) = 5; goats (g) = 7; doors (d = c + g) = 12; doors picked initially (p) = 3; doors subsequently opened (o) = 2; doors picked by switching (q) = 2. 4.3. All Possible Scenarios as Sequences of 5 Picks of Do lies in the nature of pro rent scenarios can be expected, each ee of probability. In the example at h enarios consist of 5 successive picks ther cars or goats. There are 3 initial 3) and 2 additional picks of doors by switching doo = 2) after 2 doors revealing goats have been opened 4.4. The 32 Possible Sequences of Picks To be determined first are all the possible sequences of 5 picks in which either cars (c) or goats (g) are picked. There are 32 possible sequences, as follows: 1) ccccc (picking a car at every pick); 2) ccccg (picking 4 cars and then 1 goat); 3) cccgc (picking 3 cars, then 1 goat, and finally 1 ca ccggg; 9) cgccc; 10) cgccg; 11) cgcgc; 12) gcccc; 14) gcccg; 15) gccgc; 16) gccgg; 17) cggcg; 19) cgggc; 20) cgggg; 21) gcgcc; 22) gcgcg; 23) gcggc; 24) gcggg; 25) ggccc; 26) ggccg; 27) ggcgc; 28) ggcgg; 29) gggcc; 30) gggcg; 31) ggggc; and 32) ggggg. More generally speaking, the number 32 is obtained as follows according to the theory of permutations. At 1st pick, there are only 2 possible scenarios: one picks either a car or a goat. In the 1st and 2nd picks combined, there are 4 possible scenarios: after picking a car in the 1st pick, one can pick either a car or a goat in the 2nd pick; likewise, after picking a goat in the 1st pick, one can pick either a car or goat in the 2nd pick. In other words, the mber of possible scenarios has doubled from the 1st pick to the 2nd pick from 2 or 21 to 4 or 22. It is easily seen that the number of possible scenarios will likewise double at every successive pick. Accordingly, the num- ber of possible scenarios after 5 picks will be 25 or 32. 4.5. All Possible Scenarios as Sequences of 5 Picks of Doors Hiding Cars or Goats The probability of each single pick is a fraction whose numerator is either the number of available cars or the number of available goats and whose denominator is the number of available doors. At 1st pick, all the cars, goats, and doors are still available for picking. Accordingly, in that is, 5/12, and the chance of picking a goat is g/d 12. After each pick, the denominator or the number of doors decreases by 1, from d to d − 1, and so on. The number of available doors decreases additionally when doors are opened to reveal goats. In the example at hand, the number of available doors first decreases from 12 to 9 as 3 doors are picked. The number then further de- creases to 7 when 2 doors are opened to reveal goats. Finally, the number decreases to 5 as 2 more doors are picked by switching doors. The number of available cars does not decrease when a goat is picked. Nor does the number of available goats when a car is picked. By contrast, the number of doors decreases at every pick. It follows that the probability of picking a car or a goat changes at every successive pick because at least the number of available doors, which constitutes the denominator of the probability of each pick, changes. 4.6. Conditional Probability as a Property of All Picks Preceded by Other Picks Each pick decreases the number of the available doors as well as either the number of available cars or the number of available goats. In that regard, each pick of either a car or a goat changes the probability of later picks of either a car or a goat. In ot pens in an earlier pick or earlier picks. An event whose ability is affected by what happens in is called a dependent event. Events on which other events are dependent may be called lead events. In the Monty Hall problem and its extensions, only the very 1st car picks and the 1st goat picks of sequences of picks are not dependent. An event is usually called dependent in the context of the combined probability of 2 or more events in which some events are dependent and others are not. Thus, the combined probability of picking two cars in a row is 11cd cd . The 1st pick is the lead pick. The 2nd pick is the dependent pick. Earlier picks serve as conditions of the probability of later picks. Accordingly, the general phenomenon in which the probability of a later event is changed by an earlier event from what its probability would have been without that earlier event taking place is called condi- tional probability. For exampe, throbaility of picking a car when all cars and all doors are still available is c/d. But once 1 car is picked, the number of cars and doo le pb rs both decrease by Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 250 1, the assumption being that one cannot pick the same door twice. The probability of picking a car therefore changes to 11cd. When a goat is picked in- stead of a car, the probability of picking a car changes instead to 1cd. At the same time, the probability of picking a goat chang es to 11gd . The degree to which a prior event changes the prob- ability of an event from what it would have been without that prior event can be quantified. In the example at hand, the change in probability from c/d to 11cd that results from the pick of a car corresponds to a diminution in probabilitybout 5.3%, from 5/12 to 4/11. By contrast, the change from c/d to of 7/132, or a 1cd that re- sults from of a goat corresponds to an increase in probability of 5/132, or om 5 the pick about 3.8%, fr/12 to 5/11. In sum, conditional probability is best measured or quan- tified as the degree of change between a 1st event and a 2nd event whose probability depends on the 1st event. 4.7. The General Denominator of the Equation in §4.1 It has been noted in §4.4 that there are 32 possible se- quences of 5 picks in the example at hand. Each se- quence of 5 picks comes with its own probability. The specific denominator of all 32 probabilities is the same, namely 123 4ddddod o, or in the example at hand, 1212112 212 2312 2 4, or 12 × 11 × 10 × 7 × 6. In other words, 12 × 11 × 10 × 7 × 6 is the mon dcomenominator of all 32 probabilities. The sign × r of 12 factorial, or also of 12 × 11 × 3 × 2 × 1, and that 7 × 6 is the 1st portion of 7!, or of 7 factorial, or also of 7 ppears that 12 separates the picks of doors before doors are opened from the picks of doors after doors are opened. What is the general form of the denominator? It ap- pears that the expression 12 × 11 × 10 is the 1st portion of 12!, o× 10 × 9 × 8 × 7 × 6 × 5 × 4 × 6 × 5 × 4 × 3 × 2 × 1. In fact, the two components of the denominator will always be portions of factorials. The need therefore arises to represent the two components of the denomina- tor in general forms as portions of factorials. In that regard, 12 × 11 × 10 is nothing but 12! divided by 9!, or 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 divided by 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1. The elimination of the common factor 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 yields the desired 12 × 11 × 10. Likewise, 7 × 6 is the same as 7! divided by 5!, or 7 × 6 × 5 × 4 × 3 × 2 × 1 divided by 5 × 4 × 3 × 2 × 1. The elimination of the common factor 5 × 4 × 3 × 2 × 1 yields the desired 7 × 6. In converting 12!/9! into a general form, it a is the number of doors (d) and 9 is the number of the doors (d) minus the number of doors picked before doors are opened (p), that is, d − p. Consequently, the general equivalent of specific 12!/9! is d!/(d − p)!. In converting 7!/5! into a general form, it appears that 7 is the number of doors (d) minus the number of doors picked before doors are opened (p) minus the number of opened doors (o), that is, dpo , and that 5 is the number of doors (d) minus the number of doors picked before doors are opened (p) minus the number of opened doors (o) minus the number of doors picked after doors are opened, that is, dpoq . Consequently, the general equivalent of specific 7!/5! is d pod poq !. It may be concluded that the general form of the de- nominator of the fraction that expresses the probability that one will get at least 1 car by switching doors for any number of d, , or q is as follows: c, g, p, o ! ! !! dpo d dp dpoq . 4.8. The Specific Numerators of the Probabilities nd. Each se- quence of 5 picks comes with its own probability. Each of these 5 probabilities is expressed by its own fraction and each fraction has its own numerator. The numerators befo doord. of the 32 Sequences of Picks in the Example at Hand It has been noted in §4.4 that there are 32 possible se- quences of 5 picks in the example at ha of the 5 individual probabilities of all the 32 scenarios are as follows, with × again separating the picks of doors re doors are opened from the picks of doors after s are opene 1) ccc × cc: 12 34cc ccc 2) ccc × cg: 12 3ccccgo 3) ccc × gc: 12 3cccgoc 4) ccc × gg: 12 1cccg o g o 5) ccg × cc: 123ccgcc 6) ccg × cg: 121ccgcg o 112ccgg oc 7) ccg × gc: 11ccgg og o 2 8) ccg × gg: 123cg ccc 9) cgc × cc: 12 1cg ccgo 10) cgc × cg: 112cg cgoc 11) cgc × gc: 112cgcg og o 12) cgc × gg: 123gc ccc 13) gcc × cc: 14) gcc × cg: 12 1gc ccgo 112gc cgoc 15) gcc × gc: 16) gcc × gg: 112gccg og o Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 251 17) cgg × cc: cgg 112c c 11 2g o 121o c 123g o 112c c 11 2g o 121o c 123g o 18) cgg × cg: cg gc 19) cgg × gc: cg gg 20) cgg × gg: cg ggo 21) gcg × cc: gcg 22) gcg × cg: gc gc 23) gcg × gc: gc gg 24) gcg × gg: gc ggo 25) ggc × cc: 112c c gg c 26) ggc × cg: 11 2g o 21o c ggcc 27) ggc × gc: 1ggcg 28) ggc × gg: 123g o 12 1cc 3cg o 3 gg cg o 29) ggg × cc: gg g 30) ggg × cg: 12ggg 31) ggg × gc: 12 g gggoc 4g o how each single fn the prned. The principles of condi prob- ab explicated above. Suffice it to note that the cars available for picking decreases by 1 r gets picked. And so dos the number ofe a goat gets picked. Indition, thhe goats decreases by the number of op at the 32 sequences of picks are all eq. For example, picking 5 cars in a ro (n car by switching doors, only those se 2, 16, quenrs witheorder- 32) ggg × gg: 3 12gggg o I refrain from detailing actor i oducts is obtaitional ility have been e number of th every time a cae the goats every tim ad e number of t ened doors. It is not the case th ually probablew o. 1) is naturally less probable than picking 5 goats in a row (no. 32) because there are fewer cars to pick. 4.9. The Specific Numerators of the Probabilities of the 24 Sequences of Picks That Yield at Least 1 Car in the Example at Hand In order to obtain the numerator of the probability that one will get at least 1 quences of picks in which either or both of the 2 picks made after doors have been opened yield at least 1 car can be considered. Or, the 8 sequences that yield no car need to be eliminated. They are sequences 4, 8, 1 20, 24, 28, and 32. In the list below, the 8 sequences in question have been removed. What is more, the se- ces have been reordered and so have the facto in the sequences to assimilate like to like. R ing the factors is obviously possible because multiplica- tion is commutative. But no factors have been moved across the symbol × because the factors at both sides of × belong to different picks as events. Also, the order of c and g has not been changed in the expressions of the type ccc × cc. The result of this reordering is the following 8 groups of sequences, numbered i-viii. Group i 1) ccc × cc: 12 34cc ccc Group ii 2) ccc × cg: 12 3ccccgo 3) ccc × gc: 12 3ccccgo Group iii 5) ccg × cc: 123ccgcc 9) cgc × cc: 123ccgcc 13) gcc × cc: 123ccgcc Group iv 6) ccg × cg: 121ccgcg o : 121ccgcg o 7) ccg × gc 121ccgcg o 10) cgc × cg: 121ccgcg o 11) cgc × gc: 121ccgcg o 14) gcc × cg: 121ccgcg o 15) gcc × gc: Group v cc: 112cg gcc 17) cgg × 112cg gcc 21) gcg × cc: 112cg gcc 25) ggc × cc: Group vi 11 2cg gcgo 18) cgg × cg: 11 2cg gcgo 19) cgg × gc: 11 2cg gcgo 22) gcg × cg: gc: 11 2cg gcgo 23) gcg × 11 2cg gcgo 26) ggc × cg: 11 2cg gcgo 27) ggc × gc: c: Group vii 12 1gg gcc 29) ggg × c Group viii 12 3gggcg o 30) ggg × cg: 12 3gggcg o 31) ggg × gc: what follows is to constructhe general exe probability that one will gee car by swmber of d, c, g, p, o, or qrom the 24 prtors listed above. In doing , I am de- lib explicit than might otherwise be the case ins journal in order to be more accessible anulterior design of the present effort lies afd mathematics. It is the description of the structure of human intelligence. The design of t pression for tht th itching for any nu f oducts of 5 facso erately more a mathematic d inviting. The ter all beyon Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 252 In turning the raexample at hand into a genel expres- sions need to be performed: 1) the addi- tioences of coefficients, one relating to p an on, two operati n of two sequ d the other to q, and 2) the addition of factorials. Once these two operations have been performed, it can be de- termined whether any simplifications are possible. The addition of factorials has already been discussed above. For example, a product such as 12cc c, that is, 5 × 4 × 3 in the example at hand, can first be converted into !3!cc, that is, 5!/(5 − 3)! or 5 × 4 × 3 × 2 × 1/2 × 1 in the example at hand. It can then be generalized to !!ccp. It will therefore be useful to turn first to the co which like ha sting of 1 On clos 2nd group consisting of nos. 2 and permu of the 2 doors picked, 8 groups of sequences listed in §4 efficients, which involve “the most famous of all number patterns” [7]. 4.10. The Coefficients of the Probabilities of the 24 Sequences of Picks That Yield at Least 1 Car in the Example at Hand In the list of products in §4.9, ins been as- similated to like, there are 8 groups of sequences of picks consi, 2, 3, 6, 3, 6, 1, and 2 sequences respec- tively. How can these numbers be accounted for? er inspection, it appears that they have every- thing to do with how many permutations of c and g there are in the 2 components before and after ×. For example, in the 3, the initial product is either ccc × cg or ccc × gc. Before ×, there is 1 permutation, namely ccc. After ×, there are 2 tations, namely cg and gc. Accordingly, there are 1 × 2 or 2 members in the group. In the 3rd group, there are 3 permutations before ×, namely ccg, cgc, and gcc, and 2 permutations after ×, namely cg and gc. Accordingly, there are 3 × 2 or 6 members in the group. Furthermore, the reason that there are 3 permutations before the symbol × in the 3rd group is that there are 3 picks of doors (p) before doors are opened and each of the 3 picked doors, either the 1st, the 2nd, or the 3rd, can hide the 1 goat (g) that is picked in each of the 3 se- quences in question. Also, the reason that there are 2 permutations after the symbol × is that there are 2 picks after doors are opened and each either the 1st or the 2nd, can hide the 1 pick of a goat (g) that is part of the sequences in question. The product 3 × 2 is therefore nothing but p × q. The members of the .7 all share the same sequence of picks once the factors have been reordered. In other words, there are only 8 different sequences among the 24 sequences listed in §4.7. They are as follows. Sequence i 12 34cc ccc Sequence ii 12 3ccccgo Sequence iii 123ccgcc Sequence iv 121ccgcg o Sequence v 112cg gcc Sequence vi 11 2cg gcgo Sequence vii 12 1gg gcc Sequence viii 12 3gggcg o f times that each of the 8 sequences is represented, namely 1, 2, 3, 6, 3, 61, and 2 times re- spectively, my be called the coefficient of the 8 se- qun noted above that the numbers of times in question are determined by botp and q. It ap- pears, therefore, that each sequence is characterized by tw one derived from p and the other de- rived from q. It is the product of the two coefficients that constitutes the compound coefficient of each sequence. nd coefficients in question can now be determined in terms of p and q by counting permutations of c and g before and after the symbol × in each of the 8 gr The number o , a ences. It has bee h o coefficients, The 8 compou oups of sequences. The factors 112pp and 1112ppp q found in coefficients v-viii is discussed in §4.14 when the example at hand is general- ized to yield an expression that applies to all possible cases. Coefficient i 1 × 1 = 1 (also 1 × 1 in general) Coefficient ii 1 × 2 = 2 or 1 × q Coefficient iii 3 × 1 = 3 or p × 1 Coefficient iv 3 × 2 = 6 or p × q Coefficient v 3 × 1 = 3 or 11 pp 12 Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 253 Coefficient vi 3 × 2 = 6 or 1pp 12 q Coefficient vii 1 × 1 = 1 or 11 111 12 pp p Coefficient viii 1 × 2 = 2 or 11 2 pp qq p pertaining to p exhibit the sequence 1, , ,1pp . This 1st sequence returns to 1. The coefficients pertaining equence 1, q. This 2nd se- quence is characterized by 2 properties. First, the 2nd se, and expands, each single coefficient of the 1st sece. The combined sequence 1,,11q. Second, to 1. The reason for the 2n that the 8 sequences of picks that result ioutcome of not picking a car when switchingeen removed (see §4.9 By uniting the coefficients i-viii with sequences i-viii, on pr 1 1 The coefficients to q exhibit the s nated to quen 1,p t re quence is subordi is therefo,qp the 2nd sequence does noturn re 11,, ,q q d characteristic is n the undesired doors have b). e obtains 8 products whose sum is the numerator of the obability that one will get at least 1 car by switching doors in the example at hand. The factor 1 × is explicitly expressed for transparency. Numerator part i: Sequence i with coefficient i 111 234cc ccc Numerator part ii: Sequence ii with coefficient ii 1123qccccg o Numerator part iii: Sequence iii with coefficient iii 11 23pccgcc Numerator part iv: Sequence iv with coefficient iv 12 1pqccg cgo Numerator part v: Sequence v with coefficient v 111 1 pp cggcc Numerator part vi: Sequence vi with coefficient vi 2 12 11 2 12 qcggc go 1pp Numerator part vii: Sequence vii with coefficient vii 11 11 123 pp qgggcgo 12 p It appears that, of the 8 numerator parts listed above, i an vi, and members of each of the 4 pairs of numerator pre extracted and what remains is added up, one obs 4 co tor 1 d ii share common factors, as do iii and iv, v and vii and viii. When the common factors of the 2 arts a tain mpound numerator parts. In the following list, the fac- × is again retained for transparency. Furthermore, 11 12 p Numerator part viii: Sequence viii with coefficient viii 2 1 pp gg gcc 1112pp p is the same as 1. Compound numerator part i + ii 112cc c 13 4 3cc qcgo Compound numerator part iii + iv 1 1232 1 pccg cc qcgo Compound numerator part v + vi 11 12 2 1 pp cg g c qcg 11 2c o ii Compound numerator part vii + vi 11 12 12 pp gg g p 11 3ccq cgo The sum of these 4 partial compound numerato con- stitutes the numerator of the probability that one will get at least 1 car byhe 4 compound nu- m more compactly as follows, among others because rs switching doors. T erators in question can be presented 1112pp p is the same as 1. i + ii: 12cc c 34 3cc qcgo iii + iv: 1 23 21 pc cg cc qcgo v + vi: 1 12 12 pcg g cc qcgo vii + viii: 12 13 g g ccqcg o g Before deriving a general expression applying to all Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 254 ca he specific example at hand, it will be useful to complete the example by computing the probability that it involves of getting at least 1 car by switching doors. 4.11. The Probability That One Will Get at Least 1 Car by Switching in the Example at Hand Replacing the letters in the 4 partial compound numera- toat the end of §4.10 by the pertinent num- bers and resolving the subtractions and the divisions yields the following partial numerators. ses from t rs obtained i + ii: 54321225 iii + iv: 354732 234 v + vi: 357643 243 vii + viii: 76554252 The sum of these 4 sequences is, as it happens, exactly 45,000. This is the numerator of the probability that one et at least 1 car by switching doors in the example will g is, the prob- ability of getting at least 1 car in the 3 initial picks (p)? The probability of getting at least 1 car is the same as the probability of not picking a goat 3 tims in a row in the 3 initial picks. The numerator of the probability of picking a at hand. The denominator is 12 × 11 × 10 × 7 × 6 (see §4.7) or 55,440. Consequently, the probability itself is 45,000/55,440 or about 81.2%. How does this probability compare with the probabil- ityhe car before switching, that of getting t e goat 3 times in a row is 12gg and the de- g nominator is 12ddd . The probability in question is therefore 7 × 6 × 5/12 × 11 × 10, or 7/24, or also about 29.2%. The probability of not picking a goat 3 times in a row, or also of picking at least 1 car, is therefore about 70.8%. In other words, one does somewhat increase one’s chances of picking at least 1 car when switching doors, from about 70.8% to about 81.2%, by a little over 10%. 4.12. A Key Difference between Monty Hall 1.0 and 2.0 and Monty Hall 3.0 and Higher What makes Monty Hall 3.0 much more interesting than Monty Hall 1.0 and 2.0 is t Monty Hall 1.0 and 2.0, o he following. In ne always increases one’s chances of get- tin g may ei- th some titillating variants of the expanded Monty Hall 4.13. g 1 car by switching doors when doors are opened to reveal goats [8]. But in Monty Hall 3.0 and higher, de- pending on the conditions and what the desired aim is, one’s chances of being successful by switchin er decrease or increase. A full study of these conditions exceeds the scope of the present paper. A complete un- derstanding of them should make the construction of problem possible. Some reflections follow in §6. First Generalization of the Numerator in the Example at Hand by Introducing Factorials So far, what has been obtained in regard to the example at hand is 4 compound products, the following (§4.10). i + ii: 12 34 3 ccc cc qcgo iii + iv: 1 23 21 pc cg cc qcgo v + vi: 1pcgg 12 12cc qcgo ii: vii + vi 12 13 gg g ccqcg o of these four compound products constitutes thtor of the probability that one will get at least 1 car by switching doors in the example at hand of the extended Monty Hall problem. How to pr from here? ctive thinking, there is no need for many ex- amples or many experiments to obtain the truth about a matter as there is in inductive thinking. The truth can be seen in, and generalized from, a single example. In de- riving the general truth about the probability at hand frmple at hand, the following observation can serve as a point of departure. The number of cars or goats decreases by 1 with each successive pick of 1 car or 1 goat. Accordingly, the se- qu The sum e numera oceed In dedu om the exa ences of products of factors listed above can be inter- preted as incomplete or partial factorials or snippets of factorials. For example, the sequence of factors in the product 12ccc , in this case 5 × 4 × 3, is part of the factorial c!, in this case 5 × 4 × 3 × 2 × 1, or 5!. In cases in which there are fewer cars or goats than there are picks, a factor will reduce to zero and the probability of the sequence of picks of events in question will be 0. In a next step, the partial factorial 5 × 4 × 3 can be ob- tained by dividing the complete factorial 5 × 4 × 3 × 2 × 1 by the rest of the factorial, namely 2 × 1, or 2!. In this case, 12ccc equals c! divided by 2!. However, if c were 6 and not 5, 12cc c would equal c! divided by 3!. It is therefore desirable to generalize the expression of the division of a complete factorial by a partial factorial to any c. Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 255 In that regard, it appears that the relation between the number of the complete factorial c! and the number of the partialways the same. The number of the partial factorial is always 3c because the number of the picks is always 3, however many cars there are. The divisions of complete factorials by partial factorials can therefore be generalized by expressing the number al factorial is of th et e partial factorial in its relation to the number of the complete factorial. In the case at hand, the partial factor can be expressed as 3!c and the division of the compl the partial factorial as e factorial by 3!cc. By this same procedure, umerators listed above can be converted into the following equivalents. i + ii: the 4 partial n 3! 3!! ! 3! 5!4!1! ccgo cq cc cgo iii + iv: !! 2! 1! 2! 2!1! 4! 3!2! cg pcg ccgo q ccgo v + vi: !! 1! 2! cg pcg 1! 1! 2! 3! 2!3! ccgo q ccgo vii + viii: 3! ! 4! go go !! 3! 2! 1! gc c q gc c By being converted into !3!cc, axpression cc been generalized to a certain in that it only applies when p all the terms in the equivalents listed above on when p is 3 and q is 2. The need is for con- verting the terms into expressions that apply to any p and any c. But before proceeding to the generalization to any p and any c, it is necessary to detail the general structure of coefficients. The coefficients relate to how many times eances of picks are taken. They ha been discussed provisionally in §4.10. The need at this point is for a general treatment. 4.14. The Structure of Coefficients ficients of t d in Pascal’s Arithmetical Triangle. How these nu many co- efficients there are) of a compound quantity consisting of er n, that is, how o o y can b n e such as degree. Bu 12c has t it is still specific is 3. In fact, ly apply ch of the possible seque ve already The coefhe Equation pertaining to the ex- tended Monty Hall problem exhibit the same structure as the coefficients of the power of a compound quantity that consists of two members, that is, n ab. The basic facts about this structure have been well-known for more than four centuries. They involve the numbers that are also foun mbers are obtained may be briefly reviewed below to make the present account fully self-sufficient. A particu- larly lucid and at the same time delightfully parsimoni- ous presentation of the matter at hand is Euler’s in his “Elements of Algebra” [9]. The number of the coefficients (that is, how 2 members a and b raised to the pow n ab, equals the number of the power of the com- pound quantity, that is, n, augmented by 1, or n + 1. The number n + 1 is also the number of ways in which the 2 members can be arranged in regard toften they are taken. Thus, 5 ab yields 6 coefficients, that is, the power 5 plus 1. Accordingly, there are 6 arrangements when it comes to how often the 2 members a and b of the compound quantity can be taken. One can take 5 times a and 0 times b, 4 times a and 1 time b, 3 times a and 2 times b, 2 times a and 3 times b, 1 time a and 4 times b, and 0 times a and 5 times b. If the items are multiplied, the 6 arrangements are as follows: aaaaa, aaaab, aaabb, aabbb, abbbb, and bbbbb, which can also be written as a5, a4b, a3b2, a2b3, a1b4, and b5. The 6 arrangements are the 6 main terms of the compound quantity. Each main term has its wn coefficient. The coefficient numbers (that is, what the numbers of each individual coefficient are) are determined by the number of the ways in which the 2 members of the com- pound quantite ordered in each of the arrange- ments that relate to how often they are taken. The ele- ments can be ordered in only 1 way in aaaaa. Accord- ingly, the coefficient of a5 is 1. There are 5 ways of or- dering the elements in aaaab, namely aaaab, aaaba, aa- baa, abaaa, and baaaa. Accordingly, the coefficient of a4b is 5. Along these same lines, the coefficients of a3b2, a2b3, a1b4, and b5 can be determined to be 10, 10, 5, and 1 respectively. In sum, 5 ab equals a5 + 5a4b + 10a3b2 + 10a2b3 + 5a1b4 + b5. Coefficient numbers can also be obtained as follows without having to count ways of ordering elements. If all the letters are different, as in abcde, the number of ways in which the letters can be ordered is the factorial of the number of letters, in this case 5! If 2 letters are the same, as in abcdd, 5! needs to be divided by 2! Therefore, in aaabb, 5! needs to be divided by both 3! and 2! The re- sult is 10. Furthermore, 5!/(3!2!), or (5 × 4 × 3 × 2 × 1)/ (3 × 2 × 1× 2 × 1), equals (5 × 4 × 3)/(1 × 2 × 3). The 6 coefficients 1, 5, 10, 10, 5, and 1 therefore equal 1, 5/1, (5 × 4)/(1 × 2), (5 × 4 × 3)/(1 × 2 × 3), (5 × 4 × 3 × 2)/(1 × 2 × 3 × 4), and (5 × 4 × 3 × 2 × 1)/(1 × 2 × 3 × 4 × 5) Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 256 respectively. The progression of the coefficients from 1st term to lan be gest term caneralized as follows for any power n. 112 1, ,,,, 112123 122 , 123 1 12 1 1 12 3 nnnnn n nnnnn n nnnnn n The last 2 coefficients can also be written as 12212 1 and 123 1123 nn nnn n nn . The 1st coefficient is always 1 because there is only 1 way of ordering the 1st term. The last term also equals 1 for the same reason. The coefficients involved in the extended Monty Hall problem are likewise obtained as the ways in which 2 elements can be ordered in each of the arrangements that relate to how often the 2 elements are taken. In this case, the coefficients do not equal the number of a power plus 1, but rather the number of picks of doors plus 1. The symbol a of the compound quantity corresponds to pick- ing a car; the symbol b, to not picking a car or to picking a goat. d above has only 1 coefficient. By contrast, each term of the probability sought in the extended Monty Hall problem has 2 coefficients if there are 2 events of picking more th 1st event of picking doors, that is, p. Each term of the compound quantity discusse an 1 door and therefore 1 event of switching doors. The 1st coefficient of these 2 coefficients is derived from the number of picks in the The 2nd coefficient is derived from the number of picks in the 2nd event of picking doors, that is, q. The progression of the 1st coefficient is obtained by replacing n by p in the progression listed above. The progression of the 2nd coefficient is obtained by replacing n by q in the progression listed above and leaving out the last term. The penultimate term of the progression of the coeffi- cient q therefore becomes the last. It is as follows. 12 32 1232 1 qq q qq Or also as follows. 1232 1232 1 qqqq qq q qq The reason for the removal of the last term along with its coefficient is the removal of the undesired scenarios in which 0 cars are picked in the 2nd event of picking doors. The number of the coefficients that each term has in- creases with, and is the same as, the number of events of picking more than 1 door. It also increases with, but is 1 less than, the number of events of switching doors. 4.15. The Relation between the Probability of a Sum of Partial Sequences of Picks, Either Anterior or Posterior, to the Probability of Full Sequence of Picks The quest involved in the Monty Hall problem and its extensions is first to establish both the probability of achieving an end by picking doors before doors are op on- s the Sum of the ened and the probability of achieving that same end by picking doors after doors have been opened and then to compare the two in order to determine whether, after picking doors, one improves one’s chances by switching to other doors after doors have been opened. In the example at hand, there are 32 different se- quences of picking 5 doors that lead to getting at least 1 car by switching doors (§4.10) and hence 32 different numerators of the probabilities of the sequences c ceived as single events. An example of a numerator i (1)(2) (3)(4)cc ccc . It pertains to the sequenc ich all picks are car picks. The denominator is the or all 32 sequences, namely e in wh same f 12 34ddddo co . The probability of an individual car or goat pick con- ceived as a single event is expressed as a ratio of a num- ber of available cars or goats to a number of available doors. But a sequence of picks can also be conceived as a single event. Its probability is the product of the prob- abilities that all individual picks belonging to the se- quence would have if each were conceived as a single ev probability would have bee cks, its probability, namely g/d, is independent. But when a goat p is the 2nd goat pick of a sequence, the numerator ofprob- ab ent. Many of the probabilities of individual picks in the example at hand are conditional or dependent. A prob- ability of an event is dependent if it is in part determined by what happens in a prior event. In other words, the n different if the earlier event had not taken place. For example, when a goat pick is the 1st goat pick of a sequence of pi ick its ility will be 1 g , one less goat being available be- cause of what happened in the 1st goat pick. The de- nominator will be 1d if the 2nd goat pick immedi- ately follows the first. Each of the 32 sequences of 5 picks in the example at hand consists of an anterior sequence of 3 picks before doors are opened and a posterior sequence of 2 picks after doors have been opened. The probability of either an anterior or a posterior sequence is the product of the probabilities that all individual picks belonging to the Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 257 anterior or posterior sequence would have if each were conceived as a single event. The 32 sequences of 5 picks constitute all possible cases. Furthermore, the 32 sequences are exclusive events. No 2 sequences can happen at the same time. Or, one or the other of the sequences must be the case. The sum of their probabilities is therefore 1 or 100%. The 32 sequs can be collectively evaluated in search of certain properties. In the example at hand, the first 3 picks are eved in order to single out those sequences in which on ence aluat e gets at least 1 car in those 3 pi ences of 5 picks in which th ut the pr e multiplication by 1 does not c cks. Each sequence is an event with its own probability. Moreover, the sequences are exclusive events. The prob- ability of all the sequences in which one gets at least 1 car in the first 3 picks is the sum of the probabilities of getting at least 1 car in each sequence. The 4th and 5th picks are next evaluated in order to single out those se- quences in which one gets at least 1 car in those 2 picks. The probability of all the sequ is condition is met is the sum of the probabilities of the individual sequences. But what about the probability of what happens in the 4th and 5th picks in all those sequences in which one gets at least 1 car in the first 3 picks? And what abo obability of what happens in the first 3 picks in all those sequences in which one gets at least 1 car in the 4th or 5th picks? It appears that all possible cases are consid- ered in those other picks. The probability of each case will vary depending on what happens in the remaining picks of the full sequence. But the total probability of all possible cases is 1 or 100%. It follows that, to obtain the probability of the sum of the full sequences of 5 picks that have been selected on the basis of what happens ei- ther in the anterior or in the posterior sequence of picks, one multiplies the sum of the probabilities of the anterior or the posterior sequences of picks with the total prob- ability of either the posterior or the anterior sequences of picks, which is 1. Sinc hange a number, the probability of the sum of all the full sequences of 5 picks that have been selected is the same as the probability of what happens either in the anterior or the posterior sequences alone. Consider the example at hand, in which the aim is to get at least 1 car. Once the sequences in which one gets at least 1 car in the posterior sequences have been se- lected from among the 32 sequences listed in §4.8, it is possible to evaluate the total probability of all that hap- pens in the 1st, 2nd, and 3rd picks preceding each of the selected sequences. This total probability is the sum of all the probabilities of each of the ways in which the first 3 doors can be picked. The denominator shared by all these probabilities is also the denominator of the total probability, namely 12ddd. The numerator of the total probability is the sum of the numerators of the probabilities of all 8 possible sequences of car picks and goat picks, as follows: 12cc c, 1cc g, 1cg c , 1gc c , 1cgg, 1gcg, 1 g gc, and 12gg g . These 8 sequences can be brought out in front as common factors in the selected sequences of 5 picks. Thus, as the picking of doors proceeds from the 1st pick to the 2nd pick and then on to the 3rd pick, the numerator of the probability of what happens in the first 3 picks is the sum of the 8 combinations of 3 picks just listed and the denominator is 12ddd. In nu- merical terms, the sought denominator is 12 121122 , or 12 × 11 × 10, that is, 1320. The numerator is 55 1 5255 175 7517 55 1 5771 75577172, or 543547 5 76 71771 574754 7656 75765, that is 1320. The total probability is hence 1320/1320 or 1, or also 100%. If instead the full sequences in which one gets at least 1 car in the anterior sequences are selected from among the 32 sequences listed in §4.8, the numerators of the probabilities of the anterior sequences of the full se- quences that are being selected will be the following 7: 12cc c, 1cc g, 1cgc, 1gcc , 1cg g , and 1gcg. In other words, 12gg g is not selected. The denominator of the same probabilities will always be the , namely 12ddd same compute the probability in question, a shortcut is possible (§4.11 end). The probability can be obtained by computing the probability of getting 3 goats in a row, the anterior sequces and then on the happens in the posterior ces and to resulting probabilities are compared. e anterior sequences will di elections. And of the Monty Hall problem and its exten- si To which is the only scenario in which one does not get at least 1 car, and subtracting that probability from 1 or 100%. The probability in question is about 70.8% (§4.11 end). The total probability of the posterior sequences will be 1 because all possibilities of what can happen in the posterior picks are being considered. In the Monty Hall problem, sums of full sequences of picks are selected first on the basis of what happens in en basis of what sequenhe tw Th ffer in the two sso will the posterior sequences. The purpose ons is to compare the probability of sums of anterior sequences with the probability of sums of posterior se- quences. Naturally, only picks belonging to anterior se- Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 258 quences can be considered in computing the total prob- ability of sums of anterior sequences and the same ap- plies in the case of posterior sequences. It is therefore not permissible, when generalizing the probabilities of the example at hand through the addition of factorials to unite into a single product probabilities of anterior car picks and probabilities of posterior car picks. Consider, for example, sequence iv in §4.10: 121ccgcg o. It is possible to rearrange this sequence as 12 1cccggo, bringing goat picks and car picks together. The temptation might arise to generalize 12cc c as !3!cc in an attempt to obta ab in a more general expression of the prob- ility that is sought, namely of getting at least 1 car when switching doors. But the expression !3!cc cannot be part of the expression of either an anterior probability or a posterior probability because it mixes elements of both. 4.16. Second Generalization in Terms of p and q of the Integers of the Example at Hand’s Factorialized Numerator The next step is to generalize the integers in the expres- sions at the end of §4.13 in terms of p and q. The expres- sionsr e are repeated here foase of reference, as follows. i + ii: 3! 3!! ! 3! 4! ccgo c cc iii + iv: 5! 1! qcgo !! 2! 1! 2! cg pcg c 2! 1! 4! 3!2! c go q ccgo v + vi: !!cg p 1! 2! 1! 1!2! 3! 2!3! cg cc go q ccgo vii + viii: 3! ! 4! go c cgo The sum of these expressions is the probability that one will pick at least 1 car by switching doors after doors have been opened. It will be observed that, as one moves from sequence i to sequence viii, the integers pertaining to car picks decrease whereas the integers pertaining to goat picks incr eas e. What is happening here and how does it relate to p and q? At the outset of the sequences, in sequence i, the picks are all car picks. But by the end, in sequence viii, the picks are all goat picks. In each anterior or posterior se- quence, there is a certain potential to pick cars or goats. But there is a limit to this potential. One cannot pick more cars or goats than there are picks. The maximum potential is therefore p in anterior sequences of picks, q in posterior sequences if picks, and p + q in an anterior and a posterior sequence combined. At the outset of the sequences, in sequence i, the po- te !! 3! 2! 1! gc c q g ntial to pick cars is fully exploited. In other words, nothing is taken or subtracted from the potential. By contrast, everything is taken from the potential to pick goats. However, by the end, in sequence viii, it is the potential to pick goats that is fully exploited. Or nothing is taken from that potential. It has already been noted that the numerators and de- nominators of the probabilities of sequences of car or goat picks can be considered partial factorials. These partial factorials can be presented in general fashion by dividing the full factorial by the factorial whose number is the number that follows the last number of the partial factorial. For example, in the partial factorial 12ccc , the last number is 2c. The number following 2c is 3c . The partial factorial 12ccc can therefore be presented as the full factorial c! divided by the full factorial 3!c At the same time, it is seen that the integer 3 is in fact p. After p car picks, the number of available cars has decreased by p and the numerator of the probability of picking a car in the next, 4th, pick is therefore 3c , or generally cp , because that is how many cars are still available. But this number is also the number of the full factorial by which the full factorial c! must be divided to represent the sequence 12cc c in terms of c! The sequence 12ccc can therefore be repre- sented as !3!cc By sequence viii, everything or full p is taken away from the potential p of picking cars. Accordingly, the numerator of the probability of picking cars may be presented as !!cc pp , or as !0!cc, or s, theshes be- cause no cars a also as c!/c!, which is the same as 1. In other word probability of picking cars vani re picked. In the expression !2!cc in seqii, tuence ihe in- teger 2 is only valid when p = 3. In generalizing the ex- pressions for all p, it appears that 21p. Accordingly, the expression can be generalized as !1!cc p In the expression !1!cc in sequence v, nte- ger is valid for all p, but only because sequence v is the penultimate sequence in its progression from beginning to end. As the expression 1c follows 0cp and the i 1cp and precedes t can likewise be cpp, i styled in terms of what is subtracted from p as Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 259 1cpp In the fpresentation, the integers in the se- quences found at the end of §4.13 are interpreted in terms of p and q. The expressions are presented as explicitly as possible for maximum traare also e their sum consists of the numerators of the probability of getting at least 1 car when switching doors. . ollowing nsparency. They added up becaus !c!g ii i 10! 0! cp gpp 0! 100 cp cpq ! 0! 01! cp qcp qq 0!p 01 ! go p gopqq p !! iii iv0!1! cg pcp gpp 1! 110! cp cpq 1!cp 11 ! qcp qq 1! 11! q q gopp gop p ! 1! c cpp vv i p ! 1! p g gpp 1! 10! p q 1cp cpp 1! 11! p qq cp qcpp 1! 11! q go pp p go pp pq ! 0! c cp viiviii 1 ! 1!p g gpp ! 0! cpp p q 1cp ! 1! cpp qcpp qq ! 1! gop pp gopppq q Some expressions are simplified in the following equivalent. The expressions remain unambiguous while becoming somewhat less transparent. !! ! !! 1! cp cp cq cp cpqcpq q ! 1! go goq q !! 1! 1! cg pcpgpp 1! 1! 1! 11! cp c q cp qcp qq p 1!gop p 11 ! go ppq q !! 1! cg pcppgpp 1!p 1! 1! cpp cppq 1! 11! cpp qcppqq 1! 11! go pp p gopppq q !! !1! cg cpp gppp ! ! cpp cppq ! 1! cpp qcpp qq ! 1! gop pp gopppq q This expression still reflects the values p = 3 and q = 2. There is a progression of 4 terms outside of the square brackets, 1 more than the value of p. Inside the square brackets, there is a progression of, not 3 terms or 1 more Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 260 than q, but just 2 because the 3rd term is omitted as it concerns picking no cars in the posterior sequence of picks. 4.17. Generalization of the Numerator to Any p or q In the generalization of the numerator to any p or q, there d be 1p different coefficients in regard to the coefficient p and q different coefficients in regard to the coefficient , that is, 1q minus 1 omitted coefficient, namely the last coefficient, which concerns picking no cars in the posterior sequence. In the following generalized formula, there are 5 coef- ficients in terms of p, the first 4 and the lasef- ficients in terms of q, the first 4 and the penultimate one. Th shoul q t, and 5 co e sums are infinitely expandable at every instance of the expression + ··· +. A more reduced form, still unam- biguous but a little less transparent, has been anticipated in section §4.1. In case either p or q is equal to 3 or less, there will be fewer than 5 coefficients for either p or q. !! 1cg c 0! 0! pgpp 0! 1cp c 00 ! pq 0! 00! opp goppq q g 0! 101! cp q cpq 0! 01! gopp goppq q 0! 1 12 02! cp qq cpq 0!go p p 02go ppq q ! 0! ! cp qq q cpq 12 12 303 0! 03! q q gopp gop p 12 123 2 qqqq q qq 3 2 1 q q 0! 01! qq cp cp 0! 01! gopp goppqq q !! 11! pc g cp gpp 1! 1! 111! cp cp q 1! 10! gop p goppq q 1! 111! cp q cp q 1! 11! gopp goppq q 1! 1 12 12! cp qq cp q 1! 12! gop p goppq q 1! 12 123 13 cp qq q cp q ! 1! 13! gopp goppq q 1232 1232 1 qqqq qq q qq 1! 11 cp cp qq ! 1! 11 gopp goppqqq ！ 1!! 12 2! 2! pp cg cp gpp 2! 120! cp cp q 2! 20! gopp goppq q 2! 121! cp q cp q 2! 21! gopp goppq q Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 261 2! 1 12 22! cp qq cp q 2! 22!q q gop p gop p 2! 23! cp q 12 123 qq q cp 2! 23!q gop p gop pq 12 123 2 qqqq q qq 3 2 1 q q 2! 21! qq cp cp 2! 21 q ！ gop p goppq q 12 !! 123 3! 3! pp pcg cp gpp 3 30! cp q ！ 1cp 3! 30!q q gop p gop p 3 31! cp q ！ 1 q cp 3! 31!qq gop p gop p 3 32! cp q ！ 1 12 qq cp 3! 32!q q gop p gop p 3 33! cp q ！ 12 123 qq q cp 3! 33!q q gop p gop p 12 123 2 3 2qqqq qq q qq 1 3 31! qq ！ cp cp 3! 31! go p goppqq q p 12 1 123( 1) pppppp p pp 2 !! !! cg cpp gppp ! 10! cpp cpp q ! 0! gop pp gopppq q ! 11! cpp q cpp q ! 1! gop pp gopppq q ! 1 12 2! cpp qq cpp q ! 2! gop pp gopppq q ! 12 123 3! cpp qq q cpp q ! 3! gop pp gopppq q 123 2 1232 1 qqqqqq qq q ! 1! cpp cpp qq ! 1! goppp gopppqqq 5. General Observations on Other Desired Outcomes in Monty Hall 3.0 In the special case of Monty Hall 3.0escribed in §4, the desired outcome is getting at least 1 car. But countless other outcomes may be desired. Among them are getting exactly 1 car, getting at least 2 cars, and getting exactly 2 cars, all both before and after doors are opened, as well as getting 2 cars before doors are opened and just 1 car after doors are opened. Not only the numbebut also the order of the picks can be specified. For example, the de- d r Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 262 sired outcome might be to get at least 1 car in the last door pick, and that both before and doors are opened. I hope to treat Monty Hall 3.0 more comprehen- sively elsewhere and establish the relation to the com- mon modern probability concept of hypergeometric dis- tribution. What follows are some general observations anticipating a more detailed treatment. The main observation is as is no gen- eral formula, even though certain abbreviations are pos- sible. The basic procedure is the same for all desired outcomes in Monty Hall 3.0. First, the equation in §4.1 is expanded from just the cases in which one gets at least 1 car to all possible cases, which have a probability of 1 or nt of allof car picks and after follows: There 100%. The different desired outcomes are then differe selections from the equation describing the probability possible cases. For fixed sequences goat picks, coefficients need to be dropped [10]. Suppose that the desired outcome is getting cars with every pick of a door. This is just one case of many. It is in fact a very specific case of Monty Hall 3.1. The equa- tion in §4.1 will shrink maximally. The probability of achieving the aim at hand in the anterior picks is as fol- lows. ! ! ! ! c cp d dp The probability of achieving the aim in the posterior picks is as follows. !! !! !! !! ccp cpcpq ddo dpdpoq And therefore also as follows. ! ! !! c cpq ddo !!dpdpoq The factor by which one increases or decreases one’s chances is then the following. !! !! dpoq do 6. Monty Hall 3.1: Some Reflections on Evaluating Whether Chances of Success Increase or Decrease in Monty Hall 3.0 Th nge. The challenge becomes some- what uninteresting in the expansion styled as Monty Hall 2.0 as soon as one realizes that one’s chances always increase if doors hiding goats are oened. Nothing piques human attention more than the ho of doing bet- ter or winning or the fear of doing worse or losing, let alone the combination of the two when one is not really cee. The quintessential uncertainty returns with Monty Hall 3.0, in which switching doors can result in either a de- crease or an increase of one’s chances. It is still a fact that, as in Monty Hall 1.0 and 2.0, the opening of doors always increases one’s chances. Nor will one’s chances decrease under those conditions if q is at the same time ei aller than p. The key question then is whether the increase caused by opening doors is greater or smaller than the decrease caused by diminishing the number of picks from p to q. The sys- tematic study of the mathematical conditions that deter- y Hall 3.0 in th cp cpq e original Monty Hall problem, Monty Hall 1.0, was designed as a challe p pe rtain whether one will win or los ther the same as, or larger than, p. However, one’s chances decrease when q is sm mine whether one or the other is the case in Mont may be styled provisionally as Monty Hall 3.1. Let it suffice to present in this section examples which the combined opening of doors and diminution of picks yields either an increase or a decrease in chances of getting a car. Let there be 6 doors and 1 car. Furthermore, let the number of picks decrease from p to q in that p = 2 and q = 1. One’s chances of getting the car in the 2 initial picks (p) are 11/36. That is because the chance of not picking a car in the 2 initial picks twice in succession is 5/6 × 5/6 or 25/36 and 1 – 25/36 is 11/36 or about 30.6%. The chance that the car is hiding behind one of the 4 remain- ing doors is 25/36. If 3 of the 4 remaining doors are opened to reveal a goat, the probability of 25/36 of getting the car is com- pressed into the sole door that has neither been initially picked nor opened to reveal goats. One will therefore more than double one’s chances of getting the car by switching doors even though one’s picks are reduced from 2 to 1. If 2 of the 4 remaining doors are opened to reveal a goat, the probability of 25/36 of getting the car is com- pressed into 2 doors that have neither been initially picked nor opened to reveal goats. The probability of 25/36 is distributed over those 2 doors, the chance that either door hides the car being 25/(36 × 2) or 25/72 or about 34.7%. One therefore still gains a small advantage of about 4% by switching doors. If 1 of the 4 remaining doors is opened to reveal a goat, the probability of 25/36 of getting the car is compressed into 3 doors. The probability is therefore distributed over those 3 doors. The chance that either door hides the car is erefore 25/(36 × 3) or 25/108 or about 23.1%. In this Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL Copyright © 2012 SciRes. APM 263 r of Switches of Doors (s) hance of 5/6 is com- pressed into the 3 unopened doors of those 5 other doors. re the door ori does not sw kind have in fact been done in connection with related problems. What happens if the door that one originally picked is opened? There are 2 possibilities. The 1st possibility is that the car is hiding behind that door. At this point, every consideration of probability instantly comes to naught because it is now 100% certain which door is hiding the car. There is no longer any probability prob- lem because there is no longer probability but rather cer- tainty. The 2nd possibility is that a goat is hiding behind that door. At this juncture, the situation completely changes. The door in question so far had a chance of 1/6 of hiding the car. It is now certain that it does not hide the car. The probability that it hides the car therefore drops to 0. Accordingly, the 5 other doors had so far a chance of 5/6 of hiding the car. Now it appears that these 5 doors have a chance of 100% of hiding the car. In addi- tion, 2 doors have been opened in the 1st round of open- ing doors revealing goats. Consequently, the probability that the 3 other doors hide is 100%. Each of the 3 other doors therefore has a probability of 1/3 of hiding the car. At this point, we are back at the original Monty Hall problem (Monty Hall 1.0). Now back to the generalized expression (a) below. How is it obtained? It can be obtained by generalizing the case of 2 switches of doors (s = 2) to any number of switches of doors. When there are 2 switches, there are 8 possible sequences of car picks and goat picks, as fol- lows: 1) ccc, 2) ccg, 3) cgc, 4) gcc, 5) cgg, 6) gcg, 7) ggc, case, one’s chances of getting the car by switching de- crease by between 11% and 12%. 7. Monty Hall 4.0: Additional Generalization to Any Numbe The generalization of the Monty Hall problem to any number of switches of doors (s) is styled here as Monty Hall 4.0. The following description of this generalization is limited to cases in which only 1 door is picked, as in the original Monty Hall problem (Monty Hall 1.0). The probability of getting 1 car when switching doors any number of times is as follows, with s being the num- ber of times that one switches doors, o1 being the number of doors opened to reveal goats at the 1st opening of doors, o2 the number of additional doors opened at the 2nd opening of doors, and so on (expression (a)). But before describing how this expression is obtained, it may be useful to look at the generalization at hand in a more intuitive way by means of an example. An example is as follows. Let there be 1 car and 6 doors and therefore 5 goats. An intuitive analysis is as follows. Making a diagram may be useful in following this analysis. If one picks a door, there is a chance of 1/6 of getting the car and a chance of 5/6 that the 5 other doors are hiding the car. If 2 of those 5 other doors are then opened to reveal 2 goats, the c That means that each of the 3 other doors has a chance of 5/(6 × 3) or 5/18 of hiding the car. If one switches to 1 of those 3 doors, one increases one’s chances of getting the car from 1/6 to 5/18. This also means that there is a chance of 10/18 or 5/18 + 5/18 that the other 2 of the 3 doors to which one could have switched hide the car. If 1 of those 2 other doors is now opened to reveal a goat in a 2nd round of opening doors, then the probability of 10/18 is compressed in the 1 remaining door that has been nei- ther picked nor opened. Therefore, if one switches a 2nd time, now to that 1 remaining door, one doubles one’s chances of getting the car from 5/18 to 10/18. But what happens when, switching a 2nd time, one switches back to the door that was picked first? This door tains its probability of 1/6 or 3/18. In other words, after the 2 rounds of opening doors, first 2 doors and then 1 door, there are 3 doors still to be considered: 1) ginally picked; 2) the door picked by switching; 3) the door to which one could switch by switching a 2nd time. The probabilities that these three doors hide the car are 1) 3/18 or 1/6, 2) 5/18, and 3) 10/18 respectively. This fact again illustrates the counterintuitive character of the Monty Hall problem and its extensions. For let there be 1,000,000 doors and 1 car. At 1st pick, one has a chance of 1/1,000,000 of getting the car. There is a chance of 999,999/1,000,000 that the car is hiding behind 1 of the other doors. If 999,996 doors are now opened to reveal goats, there are 3 doors left to which one could switch. They share the probability of 999,999/ 1,000,000 of hiding the car and each therefore has a probability of 999,999/3,000,000. Let us assume that one switches to 1 of these 3 doors. The chances that the car is hiding behind 1 of the other 2 doors to which one itch are therefore 2 × 999,999/3,000,000 or 1,999,998/ 3,000,000. If 1 of these 3 doors is now opened to reveal 1 goat, the 1 remaining door has a chance of 1,999,998/ 3,000,000 of hiding the car. Thus, it may strain the imagination, but it is also undeniably true, that the 3 re- maining unopened doors hold the following probabilities of hiding the car: 1) 3/3,000,000; 2) 999,999/3,000,000; 3) 1,999,998/3,000,000. Actual tests involving millions if not billions of trials, real or computer-simulated, would without any doubt confirm this fact. Similar tests of this 1 112 12 12 121 121 ss s s s ooo ds oooo (a) cddo d ddodo o L. DEPUYDT, R. D. GILL 264 and 8) ggg. The corresponding probabilities of the 8 se- quences are as follows: 1) 112 12 12 cc c ddo doo 2) 12 112 1 12 g oocc ddo doo 3) 2 112 1 12 gocc ddo doo 4) 112 1 12 gc c dd odoo 5) 112 112 1 12 g og ooc ddo doo 6) 12 112 1 12 g oogc dd odoo 7) 1 112 1 12 gogc dd odoo 8) 112 112 12 12 g og oog ddo doo The desired outcome, getting a car after two switches n sequences 1), 3), 4), and 7). The probability of getting a car after two switches of doors is therefore the sum of the 4 probabilities 1), 3), 4), and 7). The common denominator of this probability is as fol- 12 2do o (b) sams s and 1 therefore of doors, is achieved i lows. 1 1ddo And e a considering that 2 is the the same as 1 s , this expression can be rewritten as follows. 11 1 s ss dd sodoo (c) ), oe can derive the fol- number of switches (s). 112123 2 1 12 3 ss dodo odooo o o (d) as (e), without too been so abbreviated in expression (a) anticipated above. 112 1 12 s By n iny extending both (b) and (c lowg expression applying to an 123 123 1 1 ss ds ooo dsoo ooo Expression (d) can be abbreviated great loss of transparency, and it has d 12 s s ddodo o o o (e) ility at hand is the fol- sequences 1), 3), 4), and 7) above. dsoo numerator of the probab lowing sum of 4 terms found in The 1 1 12 1 11 ccccg oc gc cggoc This sum can be rewritten as follows. 1 1 12 11 1 cccccg occg cg go And therefore also in successive steps as follows. 11 11 11 11 1 1 1 12 111 12 11 12 2 122 12 12 12 ccccgoc gggo cccgo gcgo cccg ogcgo ccdo gdo cc gdo cc gdo cd do (f) And considering that 2 is the same as s and 1 therefore the same as 1 s , expression (f) can be rewritten as fol- lows. 1 1s cdsds o (g) By extending (f) and (g), one can derive the following expression of the numerato switches (s). r applying to any number of 112 123 2 122 1 123 1ss ss cddo do o dsoooo dsooo o (h) Expression (h) can be abbreviated as (i), without too great loss of transparency, and it has been so abbreviated in expression (a) anticipated above. 112 121 12 3 ss cddo do o dsoo o (i) In the specific case of the above example featuri car, 6 doors, and 2 switches of doors, the gene si ng 1 ral expres- on assumes the following form. 1 12cddo 11 2 12dd odo o Entering the relevant integers, one obtains the prob- ability already given above. 16162210 ’s chances of get- ting a car after s switches is therefore as follows. 6 6126221 18 The chances of getting a car at the first door pick is c/d. The factor by which one increases one Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 265 1 2121 12 ss 112 1 1 12 s s dd s dod d g a car from 1 switch to s switches is therefore the following. od oo o oo soo oo The factor by which one increases one’s chances of gettin 112 23 121 1212312 1 23 ss s s dso oo doodooodsoo oo Finally, in Monty Hall 3.0, when more than 1 door is pi re are 2 ru oq. Doors are switched only 1 time (s = 1). If more than 1 d or is picked at each pick and doors are switched more than 1 tim fr 8. Back to Boole. By Richard D. Gill 8.1. Summary I comment on Leo Depuydt’s recent work on applying Boole’s work in probability theory to the Monty Hall problem. In particular, I compare Boole’s notation and conventional modern probability notation, discuss mod- ern computational tools, and make some comments on Boole’s position that probability theory belongs to the laws of thought. oole’s work on probability theory stands on an equal intellectual level to his work on l tended by him to be seen as an integral p has largely been forgotten. Now that three half centuries have gone by and probability theory has flourished, fol- lowing different routes, his work is harder than ever to re find his way into Boole’s way of thinking. In this paper and its predecessor Leo tackles a number of variants of the Monty Hall problem, showing how oole’s approach leads to solutions despite ever increase- ists. I emphasize ilists con- one could better say, ill posed. laim, influential writers of the early as dependence should be assumed. In modern day terms, Boole fitted judiciously chosen log linear models to the data, judicious higher order interactions about which there was no in- fo are represented by graphs and the same graphs used as fdation for graph-theoretic based computations. However, I do not know if Depuydt is also going to “a Boole indeed saw probability theory as part of the laws of thought. His probabilities are subjective degrees of belief, their numerical values follow logically from con- si of in- difference, but using indifference not just to specify probabilities but also to specify probability structures. However, so far, we are considering here problems where all probabilities are completely specified and where a frequentist (objectivist) and a Bayesian (subject- force unique values of probabilities from “equally likely, by symmetry” arguments. rn day notation by means of a simple (mathematical) example. Consider four events which can sequence. For instance, the results dodoo cked both before and after doors are opened, the ns of coefficients, 1 for p and 1 fr o e, there will be more than 2 coefficients. I refrain om entering into detail at this time. 8.2. Introduction George B ogic, and was in- art thereof, yet it ad. It is impressive that Leo Depuydt has been able to tivist) approach will give the same probability values, since in either approach the symmetries of the problem B ing complexity. From the point of view of a present day professional mathematician, my first questions were: are the answers correct? Is Boole’s probability different from present day probability? The answers so far are yes: the answers are correct, and no, Boole (and with him Depuydt) is using the same probability rules as present day probabil so far because Boole also claimed to be able to solve probability problems which modern day probab sider insoluble, or perhaps Because of this c twentieth century such as Keynes dismissed Boole’s work completely, and that hastened its progress into limbo. As Miller (2009) points out [9], however, Boole’s solution was meaningful and complete, and based on adding an 8.3. Notation It is easiest to explain the difference between Boole’s notation and mode sumption that in absence of further information, and in particular, with no logical dependencies, an appropriate higher level of conditional statistical in ly dropping rmation anyway. This connects to modern develop- ments in graphical models (also known as Bayes nets), another development which Boole would have appreci- ated, in which probability models oun uthorize” this particular, more controversial part, of Boole’s thinking. deration of information (known and unknown). He stood here full square in the nineteenth century tradition of Laplace, deriving probabilities from the principle occur, or not occur, in of a first pick of a door, a second pick, and so on. Let me denote the events as A, B, C, D (capital letters early in the alphabet, according to present day conventions). The modern view of probability theory is that we may consider these events equally well as subsets of a set “ele- mentary outcomes.” The event A is identified with the set of all elementary outcomes for which A does indeed happen. Probabilities are assigned to subsets of , and set theoretic operations turn out to correspond to g events. For example, the logical constructions involvin Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 266 event that both A and B occur corresponds to the outcome of the probability experiment, , being both a member of the subset A and the subset B. Thus the prob- ability of A and B happening is identified with PA B, where P is a mapping from subsets of to numbers between zero and one. Subsets A, B, etc., are often called “compound events.” Provided however we are careful with language, the words “elementary” and “compound” in the two contexts “elementary outcomes ” and “compound events A,” are superfluous. But it also does no harm to add them. The elementary outcomes correspond to the most fine-grained, most detailed, description of what actually happened. Compound events correspond to coarse- grained descriptions, by which many alternative “micro- scopic” ways according to which the same “macroscopic” phenomenon can come about are all grouped together. Whatever probabilities are supposed to mean (whether relative frequencies in the long run of many repetitions, or whether degrees of belief as measured by fair betting odds), everyone agrees that if two events can never hap- pen together, the probability that either occurs is equal to the sum of their probabilities; that certainty corresponds with probability one; and that all probabilities are greater than or equal to zero. Converting these minimal properties into the language of set theory, we obtain the now familiar axioms: 1P, 0PA for all A, AB implies PA BPAPB . Finally we add as a definition of conditional probability of A given B, as long as 0PB: P ABP ABP B . From these minimal properties one can derive the fol- lowing chain rule: and and and andand and. PA B CD PAPBAPCA BPDA BC (1) However, the alert reader will have noticed that I am mixing the language of logic and the language of ele- mentary set theory in this equation, and I do that deliber- ately, in order to point out an important ambiguity in the translation from logic to set theory. The interpretation of the left hand side is obvious: I could have written (should have written!), of course, PA BCD . The right hand side certainly makes sense, and indeed the statement is true, if I do the corre- on that side; for instance, the last sponding substitutions term should be A C. However there is an Let me explain. Suppose start with a probability PD B I alternative substitution, more clumsily expressed in set theoretic language, but equally meaningful from the point of view of natural language. The correctness of this al- ternative interpretation is actually a theorem. measure P. Next I pick some event B with positive prob- ability, and compute new probabilities B PA PAB for every event A. Theorem 1: the conditional probability measure A P also satisfies the axioms of probability theory; Theorem 2 (principle of repeated conditioning): B AAB PP . This is not just empty formalism, it tells us something very important: conditioning in turn on any number of events gives end results which do not depend on the order in which we take them, and is not changed by grouping them into a smaller number of events by using the rule P(A|B and C) = also P AB C. It shows ue transition between the language of logic and the language of sets is very smooth indeed s that th . For B does fi Boole has no use for the language of sets. It was not even yet invented: his supporter and contemporary John Venn was one of those who pioneered its use; indeed, its use in probability theory.oole, the language of logic ne both for events and for probabilities of events. Defining the event E as “A and B and C and D,” Boole writes the definition of the event E as logical relatione abcd, (2) and then rewrites Equation (1), a relation between prob- abilities and conditional probabilities, with the very same sequence of symbols: numerical relation between probabilitieseabcd, (3) Even though Equation (3) is to be interpreted numeri- cally as a relation between probabilities, the rules of al- gebra have to be handled with very great care. The exact sequence of probabilities abcd corresponds to a specified sequence of events A, B, C, D and there is a logic to this sequence: typically this will be their temporal ordering. The valud tomerical ve c, f n front of events a and b. Event D might be certain in some context, impossible in other. Tced e assigne the nuariablor in- stance, depends on the context, on the presence i he preing events A, B and C could switch the probability d to 1, or to 0. This is what Depuydt calls the digital nature of probabilities. One of the fruits of the digital revolution has been statis- tical computing and computer algebra. Looking at huge tables of probabilities in sections 4.1 and 4.17, reader may worry that perhaps some typesetting error has co nother opportunity fo 8.4. Computations the the rrupted one of the formulae. If the reader actually wanted to use those formulae to do numerical computa- tions, he or she might want a computer do those compu- tations. But then the typeset symbols have to be translated into lines of computer code, which is a r errors to creep in. Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 267 I have verified that it is in principle possible to repro- sing computer algebra. Let the com- puter do the painstaking, repetitive task of applying sim- ple rules of transformations of formulas! Let the computer typeset the page let the computer also generate computer code for implementation to specific cases! Then the reader need only check the programs or scripts: do they implement Boole’s logic of probability? There are two levels involved here. The prob duce these tables u s in the journal, lem should be ich the computer ows. Anyone who understands the la m must be ch day for a myriad asks also provide external consistency checks whene answer can be got by different m Sage (http: publicly available, and an orithms which it uses are public; the scientist can check th them by new algorithms of their own. of a particular generalized Monty Hall problem, count outcomes of different kinds, in o to statistically estimate the probabilities which can in pri as cer- a short time Sage has become flexible. Like natural languages, e of ball-picking in phase one. culture, and like mathematics itself, these systems evolve through highly effective “crowd-sourcing.” 8.5. Alternative Approach Depuydt goes back to first principles and determines the probabilities of all possible elementary outcomes of his Monty Hall games: any particular sequence of picks of doors. Now, it is possible to group some of the picks to- gether, producing a coarser level of description, but one in w described in a high level formal language which translates line by line Depuydt’s verbal descriptions of what he is doing into a language wh algebra system kn hich (a) the components of the coarser description cor- respond to familiar probability models, and (b) the coarser description is fine enough to still allow specification of the compound events of interest. In Monty Hall 3.0, such a coarser description is possible at the level of phases. Recall that in this game, c doors hide cars, g doors hide goats, d = c + g is the total number of doors. The player first picks p doors. The host then opens q doors, revealing goats. The player may now switch to another r doors. The hosts’ possibilities are delimited by how many cars are hidden by the player’s first p picks. Call this number x. We can now write down the joint probability of x cars being behind the player’s first p picks, and y cars being behind the player’s second r picks, as follows. Both phases correspond to a traditional “sampling without replacement” situation, picking balls from vases, where the composition of the vase at phase two is determined by the outcom nguage can verify that it is “the same thing.” The i- plementation of the computer algebra system ecked by specialists, though users who use it day by of t ver the eans. I would like especially to draw the reader’s attention to two powerful tools, both of them completely free (both in the sense of “free beer” and in the sense of “free speech”): the statistical language R (http://R-project.org) and the computer algebra system//sagemath.org). The freedom as in free speech is the fact that the computer code of both R and Sage itself are yone is allowed not only to look at it but also to modify it, repackage it, and even to sell it, as long as their modi- fications preserve the same freedoms. Sage allows one to instruct the computer to perform algebraic formula manipulations according to specified rules. Boole would have appreciated that. Unlike com- mercial tools like Mathematica, the alg Suppose a vase contains R red balls and B blue balls, let N = R + B be the total number of balls in the vase. Sup- pose n balls are picked at random from vase, without replacement, and completely at random. Define the bi- nomial coefficient !! ! n x Cnxnx, the number of ways to choose x objects from a collection of n. In spoken mathematics, one says “n choose x” instead of “C super- script n subscript x.” Let r be the number of red balls in the sample of n, and define bnr to be the number of blue balls. It turns out that the probabilit em, even replace R is a statistical computing tool. One thing which is extremely easy with R is to run a computer simulation of millions of repetitions y to find exactly r re rder d balls is h(r; n, R, N) = R BN rbn CCC. The fact that these so-called hypergeometric probabilities must add up to one as one adds over all possible values of r is called the Chu-Vandermonde identity in combinatorics, going back to Chu Shi-Chieh, 1303, and Alexandre-Théophile Van- dermonde, 1772. One can say that Depu nciple be computed algebraically. Both these systems are widely used in academia, in teaching, in industry; they have huge followings and be- cause of their open nature, additions have been written by users from all kinds of application fields which any- one else can also freely use. The user communities with their internet fora and mailing lists and so on, allow both the new user and the expert to get advice from fellow users all over the world, often extremely rapidly and ef- fectively. R can even be used from Sage—one of the design philosophies of Sage is to use existing tools, so as not to waste time re-engineering wheels. This h ydt has derived a “two-level” generalization of this identity from first prin- ciples, following Boole’s methodology. Now if among the first p doors chosen by the player exactly x doors hide cars, then at the second stage, when there are dpq doors left from which the player may choose r doors, a further q doors already having been op tainly paid off, since in extremely powerful and ened revealing goats, exactly cx of those doors hide cars, and g px q hide goats. This tells us that the probability that the player’s first p picks hide x cars and Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 268 his second r picks hide y cars is ;,,;,,hxpcdhyrcxdp q. This gives an alternative way to check the results of this paper. 9. Empirical Definition of Mathematics, in Boole’s Footsteps, as a Cognitive Event on the Deepest Level 9.1. Where Is Mathematics? The question that is at the center of the present section is as follows: What is mathematics? The answer to this iderable interest. No endeavor of the cessful. Evidently, the een basically two diametric- mathematics e may interpre inside t ught to co said about that ca rally, number when it comes to em- pirical observation, numbers can ly be observed and th ing to the brain. Mathematics is something that the brain does. 9.2 This to pursue a line of inquiry a half ago but fairly question is of cons ca ca ca human intellect has been more suc question has occupied many, many minds over the centu- ries. The literature on the subject is massive. But even the most cursory review of what has been done readily reveals that the question can hardly be considered an- swered. There has been no lack of attempts to provide an answer. However, the proposed answers seem often ir- reconcilable and can even be diametrically opposed. In order to define mathematics, one needs to be able to observe it. A second question therefore presents itself, as follows: Where is mathematics? In other words, where n one find mathematics so that one can take a look at it and analyze it in order to determine what it really is? It appears that the answer to the seemingly simple question as to where mathematics is has perhaps been the greatest point of controversy in the discussion of what mathematics is. There have b lly opposed answers to the question where mathemat- ics is. Some believe that mathematics is something inside the head. Others believe that it is something outside the head. Whereas many believe with Kurt Gödel that num- bers exist independently of the human mind, many others like L.E.J. Brouwer are convinced that numbers are a creation of the human mind. Could both be right at the same time? The position that I will adhere to is that n only be empirically observed as something that is inside the head. This position in no way involves a denial of the notion that mathematics is something outside the head. Clearly, when applied to reality outside the head, mathematics works. Somt this as proof that mathematics is also something outside the head. Then again, the totality of human experience of reality outside the head is how the brain perceives and processes this reality through the senses he head. This percep- tion is itself 100% brain activity. Therefore, the analysis of the human experience onsist in the final re- sort, on the deepest level, of the analysis of brain activity. And one component of the brain activity that constitutes the human experience is mathematics. In that regard, the question as to whether mathematics is also something outside the brain is to some extent moot because mathe- matics cannot be empirically observed in that capacity anyhow, so there is hardly anything to be pacity. To some extent, Gödel’s position and Brouwer’s posi- tion are not in opposition. There is nothing that contra- dicts the notion that numbers are something that is both something outside the head and something inside the head. Natus would inhabit different medi- ums inside the head and outside the head, physical reality and brain mass respectively. But on erefore also analyzed as an activity of the brain or an event happen . Resuming an Abandoned Line of Inquiry e aim of what follows that was initiated about a century and soon completely abandoned and ever since entirely dis- regarded. This line of inquiry is, I believe, worthy of be- ing resumed. It appears to me that it can lead to a final definition of mathematics and its foundations. The ini- tiator of the line of inquiry in question was George Boole, first in his The Mathematical Analysis of Logic (1847) [13-16], but then above all in his An Investigation of the Laws of Thought (1854), which may be regarded as the Magna Charta of the digital age [17,18]. The principal follower of Boole was John Venn in his Symbolic Logic (second edition, 1894) [19]. Whitehead notes that Venn gave “thorough consistency to Boole’s ideas and notation, with the slightest possible change” [20] and, more re- cently, Styazkhin observed that Venn “revealed the es- sence of the secret of success of Boole’s procedures” [21]. The Digital Age owes an extraordinary debt to Boole and to the digital mathematics that he created. Digital mathematics is a type of mathematics that is distinct from the more familiar type of mathematics, quantitative mathematics (to which Boole also made significant con- tributions, for example by his work on differential Equa- tions). But digital mathematics is in the end just as mathematical as quantitative mathematics. Clearly, a line of inquiry initiated by Boole has proved to be successful. Little did Boole know to which uses his digital mathe- matics would be put when he wrote in 1847, “It would be premature to speak of the value which this method may possess as an instrument of scientific investigation” [22]. 9.3. Probability Theory as an Ulterior Aim of Said Line of Inquiry It is not clear to which extent Boole, when initiating the line of inquiry that ultimately spawned the Digital Age, Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 269 had something like computer science in mind as an ulte- rior aim. In fact, in his Laws of Thought, digital mathe- matics is clearly subordinated to an ulterior aim of an entirely different kind, namely making classical prob- ability theory complete. It is not entirely certain whether Boole had this rela- tion of subordination to probability theory in mind as soon as he began working on digital mathematics. There is no mention of probability theory in his The Mathe- matical Analysis of Logic of 1847, in which he first es- tablished his digital mathematics. But in Laws of Thought, digital mathematics is clearly styled as serving the aims of probability theory, as appears from the second part of the book’s long title, (An Investigation of the Laws of Th ght and Language as an Ulterior Aim of Said Line of Inquiry of Boole’s Work on Logic and Probability Boole of giving his 1854 book the wrong title. He be- dealght,” because “the question reflec- g the p cates with another through thought and la same ought), on Which Are Founded the Mathematical Theories of Logic and Probabilities. A great irony relating to Boole’s legacy is that his work on probability theory has been, with one or two exceptions [23], completely disregarded, almost entirely bypassed by the field. In the planned article mentioned at the end of §1, I intend to confirm that Boole’s probability theory does what it claims to do, make classical probabil- ity theory complete, and how it does so. 9.4. Rational Thou But there seems to be more to the ulterior aims Boole’s digital mathematics than statements about prob- ability theory. Boole’s Laws of Thought and many of his other works on logic and probability, both published and unpublished, are replete with references to the nature of human thought in as far as thought is rational. The ques- tion as to whether he aimed to determine what is going on in one’s head when one thinks rationally is investi- gated below. In any event, like Boole’s ideas on probability theory, this component too of his line of inquiry appears to have fallen by the wayside. Whereas the forthcoming article mentioned at the end of §1 is an attempt to validate Boole’s line of inquiry in relation to probability theory, what follows is an attempt to resume and extend this same line of inquiry as it relates to the deepest founda- tions of rational human thought and mathematics. 9.5. Is Mathematical or Cognitive in Nature? 9.5.1. Modern Perception of Boole’s Work on Logic as Strictly Mathematical When one reads Boole’s writings on logic and probabil- ity, the following question easily arises: Is Boole doing mathematics or is he trying to determine how people think rationally? In other words, is he describing the mathematical structure of reality or is he trying to tell us what is going on in people’s heads when they think ra- tionally? Boole’s contributions, to the extent that they have proved lasting, are now universally perceived as belonging to the realm of mathematics. Boolean algebra is after all ubiquitous. Bertrand Russell even accused lieved that Boole was “mistaken in supposing that he was ing with the laws of thou how people actually think was quite irrelevant to him” [24-26]. Taking into consideration how people think while practicing mathematics is sometimes called psy- chologism, which some seem to regard as a bad word. 9.5.2. Statements to the Contrary in Boole’s Writings There are abundant indications in Boole’s work that leave no doubt that how people think, at least as far as rational thought and language is concerned, was very much on his mind. In the Preface to the earlier Mathe- matical Analysis of Logic (1847), he states that he is not concerned with “quantity,” but with “facts of another order which have their abode in the constitution of the Mind” [27]. In the first statement following the Preface to the later Laws of Thought (1854), he announces [28]: “The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed.” How can such statements, when taken at face value, not pertain to what is going on inside the heads of people —notwithstanding attempts to soften their impact, per- haps to protect Boole from the charge of psychologism? [29]. Two possible reasons for resisting the notion that Boole could have been aiming to establish how the brain works are as follows. First, mixing Boole’s mathematical results with tions on the nature of thought might be seen as affectin urity and objectivity of the former. Second, at the present time, it remains still basically unknown—let alone that it was in Boole’s time—how the brain produces rational thought and language in bio- chemical terms, that is, which activities of neurons and synapses are responsible. So how could anyone have anything to say about how the biological brain reasons? The first objection is addressed below. In regard to the second objection, I have noted elsewhere that, as one brain communi nguage, all communications 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 everything that is essential to the structure of rational thought and language must be conveyed in sound waves or light beams that travel from mouth to ear or from page to eye. In that sense, the structure of rational thought and language is empirically accessible. The Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 270 structure ought to be present inside the brain, even if ting a different medium. in- habi sa is a good illustration of this definition. It seems otherwise quite tempting to in- a prope e the brain. As it happens, that the Brain d in the end avoid de ite of this assumption is that there is something more to reality than what is perceived through the senses. uld be. Relig ut that something 9.5.3. Boole’s Own Perception of His Work on Logic as Mathemati cal While there can be no doubt that how the brain thinks is somehow a prominent concern in Boole’s writings, there are also plenty of statements in his writings that leave no doubt that he is firmly convinced that what he is doing when he is studying logic and probability is mathematics. He states, for example, that “the ultimate laws of Logic are mathematical in their form” [30]. As one tries to assess what exactly it is that Boole is trying to do, the impression gradually imposes itself and becomes inescapable that he is writing both about how the mind thinks and about mathematics. There are just too many categorical statements in his work that posi- tively point to both. At this juncture, there is the possibil- ity of assuming that there is something deeply confusing and contradictory in Boole’s work. One might seek to resolve the possible contradiction by discarding either the cognitive facet or the mathematical facet of Boole’s work as invalid. In choosing to reject either of the two, the easier choice would seem to be the cognitive facet. The mathematical facet has more than proved itself by appli- cations in modern computer science. Then again, it is difficult to overlook the many pas- sages that concern how the mind thinks. Consider his analysis of the syllogism, which does not supplant Aris- totle’s analysis but rather completes it. It seems easy for all to agree that we must think according to the rules of the syllogism if we are to reason correctly. And more generally, it is easy to convince oneself that what Boole says about how the mind thinks rings true. There just seems to be more to Boole’s writings than just mathe- matics. 9.5.4. Could Boole’s Work on Logic Be Both Mathematical and Cognitive? The question arises: Could Boole have been doing both at the same time, producing mathematics and describing mental faculties? The following statement by Boole clearly indicates that his approach is at the same time mathematical and cognitive. What he sets out to discover is the mathematical structure of rational cognition [31]: “The laws we have to examine are the laws of one of the most important of our mental faculties. The mathe- matics we have to construct are the mathematics of the human intellect.” The present discussion has reached a critical juncture. It needs to be decided whether the cognitive facet of Boole’s line of inquiry should be pursued or dropped altogether. The validity of Boole’s digital mathematics ys something about the overall soundness of his think- ing. It can serve as an argument in favor of resuming the cognitive facet of the same general line of inquiry. In resuming the cognitive facet, the concept described by Boole as the “mathematics of the human intellect” cited above will serve as a point of departure. What can possibly be meant by this concept? It would seem that it places mathematics somehow inside the human intellect. The way in which the concept will be interpreted in what follows is that mathematics is in essence a property and an activity of the brain. Mathematics is best defined as something that the brain does. In a planned article, I hope to show that probability theory terpret mathematics as exactly the opposite, namely as rty of reality outsid very notion will also be assigned a place in the definition of mathematics as something that the brain does. Mean- while, the principal consideration that leads to the defini- tion of mathematics as an activity of the brain is pre- sented in the next section. 9.6. Mathematics as an Activity of The brain is evidently the most complex structure in the universe. It consists of billions of neurons and trillions of synapses. Still, it seems just as evident that the brain is a biological mass that is limited in size. There is only so much of it and no more. The following working hypothe- sis therefore seems to impose itself. The time will come when it will be possible to record everything that the brain does as it happens, presumably with the aid of su- percomputers or the like. The opposite of this hypothesis is that a certain part of the brain will be forever inacces- sible. But what could such an inaccessible part consist of? If everything in the brain is atoms and molecules and the like, then no activity in the brain shoul tection, one would think. Another basic assumption is that the totality of human existence as we know it consists of how the brain per- ceives reality outside itself through the senses. There are many more senses than the classic five, including sensing the effects of the instincts with which the brain comes equipped at birth. In addition, perceptions received through the senses can be recombined in certain ways inside the brain. Dreams are one type of recombination. The oppos It is difficult to see what that something more co ion makes certain assumptions abo more. But then, it is impossible to make everyone agree on what that something more is and the assumptions of religion are beyond scientific verification anyhow. Once it is possible to record everything the brain does in its entirety, part of what is recorded will be the brain’s Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 271 knowledge and practice of mathematics. It should be po rded brain activity? r understanding. The brain edge form g mathematics. It is an im- po rva- tio itself subject, without its being also given to it to understand their - gree r fitness for their end, as com- ssible to observe exactly what the brain does when it engages in mathematics and how it starts up mathematic- cal knowledge. The key question arises: Is there more to mathematics than reco 9.7. The Brain as the Final Frontier: Towards a New Empiricism If the totality of the human experience consists of how the brain engages what is outside itself, then nothing that does not have some kind of imprint in the brain can mean anything to the brain. In assessing what is outside itself, the brain only has itself, as it were, to sort things out. And by itself is meant a complex and very large but ul- timately limited and fully definable amount of activity of neurons and synapses and the like. At first sight, it would seem as if mathematics is a property of reality in which the brain occasionally par- ticipates. Mathematics seems like a sacred code inscribed in the book of nature. But all that the brain can ultimately know about this code is the details of its own participa- tion. And the details of this participation consist one hundred percent of brain activity. Therefore, if one truly wants to understand what mathematics is, then all one has as an object of study is the participation itself as brain activity. It is understandable that there may be a desire for more than just that. But the brain can hardly step outside itself, as it were. It is fully limited to its own activity and powers, and to the study of this activity and these powers in a search fo activity does not only include mathematical knowl and reasoning, but also the act of perception in the of signals reaching the brain from outside through the senses. Needless to say, once it is possible to observe all this brain activity, it will also be possible for this very act of observing brain activity to be itself observed, include- ing by the person whose brain activity is being observed. It is a bit like a snake biting its own tail. But what about the ever attractive notion that mathe- matics is a property of nature outside the brain? Nothing is more tempting than to subscribe to this assumption. In fact, I believe that there is nothing wrong with assuming that reality exhibits a structure that may be called mathematical and that this structure is somehow the ori- gin of a certain type of brain activity that may be de- scribed as knowing and doin ssible to avoid assumption under which everyone ef- fectively operates. One way of looking at the matter is as follows. It is not because there is no final verification of this assumption that the assumption should be rejected. The assumption receives abundant support from the fact that mathematics works. When mathematical knowledge is acquired and this knowledge is then returned to reality outside the brain by being applied correctly, as in build- ing a bridge, the application will typically work, that is, the bridge will not collapse. But ultimately, mathematics can only be observed to the extent that it can be seen at work in the brain. Reality is experienced entirely in terms of how the brain engages what is outside itself through the senses. The scientific observation and analysis of this experience therefore ultimately needs to be the obse n of the brain. And that also applies to mathematics as one type of reality. Anything else is beyond human knowledge. It is not possible to look behind the curtain, as it were, to establish why the brain is the way it is. Along these lines, Boole’s writes in somewhat Latinate English, “It may, perhaps, be permitted to the mind to attain a knowledge of the laws to which it is ground and origin, or even, except in a very limited de , to comprehend thei pared with other and conceivable systems of law” [32]. Because the assumption that the structure of reality outside the brain is mathematical is just an assumption, it is not possible to probe the deeper roots of this presumed structure. There is of course nothing that prevents anyone from engaging in speculation to any degree. It is likewise possible to speculate without restrictions about other possible types of realities in which other possible types of mathematics apply. Knowledge is ultimately a process of assimilation in which the brain assimilates to reality outside itself. For example, to find one’s way through the streets of a city without consulting a street map, the brain needs to ac- quire something of the structure of the layout of the city’s streets and in that sense become a little like that layout. But it is reasonable to assume that, in the process of assimilation, there needs to be something to assimilate to. Therefore, if part of the assimilation is mathematical, there is presumably something mathematical in reality outside the brain to which the brain assimilates. The fact that the knowledge of mathematics is stored in the books of a mathematics library may also seem to suggest that mathematics is something outside the head and hence first and foremost a property of nature, with its reflection inside the head being somehow secondary. However, the books in question are nothing more than paper and ink until an active brain reads and studies them. In that sense, a tree does not fall in the forest if there is no one there to hear it. The mathematics in a book is not mathematics if it is not actively engaged by a thinking brain. 9.8. Conclusion It is possible to reconcile as complimentary the view ad- hered to by someone like Brouwer that mathematics is Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL 272 something inside the head and the view adhered to by someone like Gödel that mathematics is something out- side the head. In other words, the two views do not con- tradict one another. However, that mathematics is some- thing outside the head is only an assumption. But it is an assumption that is hard to deny. So to some degree Gödel’s view can be recognized. Still, it is only as some- thing inside the head that mathematics can be truly ob- served and therefore become the subject of empirical inquiry once the secrets of the brain are unlocked. In that regard, the cognitive approach is the only one that offers a systematic path of scientific investigation. I hope to apply the cognitive approach in planned papers, begin- ning with the branch of mathematics called probability theory. It will be useful to formulate the foundations of probability theory fully in cognitive fashion. 10. Acknowledgements When the present article was essentially complete, a for- tunate set of circumstances brought the author in contact with Dr. Richard D. Gill, Professor of Mathematical Sta- tistics at the Mathematisch Instituut of Leiden University in the Netherlands, who has contributed a number of studies to the analysis of the Monty Hall problem. An interesting exchange of ideas ensued about all sorts of facets of the expanded Monty Hall problem and about the contents of the present article. I personally profited much from this exchange. One result of the exchange was the decision to include, for the benefit of a somewhat more interdisciplinary audience, an appendix by Professor Gill (see §8). The design of this appendix is to provide addi- tional context by building a bridge to modern probability theory in its conventional notation and by pointing to the benefits of certain interesting and relevant tools of com- putation now available on the Internet. A more detailed and in-depth description of the common concept of the hypergeometric distribution in its relation to the contents of the present article remains desirable and will need to be postponed to future papers. The present collaboration is meant as a first step conceived and executed on short notice, an exploratory effort that probes what is possible in terms of interdisciplinary projects spanning both the humanities and the sciences. I am grateful to Professor Gill for his willingness to make this much appreciated contribution. As regards the interdisciplinary nature of the larger project of whose mathematical branch this article is part, one ulterior aim is to promote the perfect unity of nota- tion of Boole’s algebra and the complete unity of human intelligence that it suggests, in that the notation can be applied to the following multiple facets of human intelli- gence, each illustrated here by one expression. 1) “The sun shines” (Language, Level of the Things). 2) “When the sun shines, I go to the beach.” (Lan- guage, Level of the Events). 3) “Humans are mortal. Socrates is a human being. Therefore, Socrates is mortal.” (Logic, but not just with three statements, as in the present example, but with any number of statements). 4) A quadratic equation (Quantitative mathematics). all problem (Digital mathematics, in 5) The Monty H addition to quantitative mathematics). Finally, I also thank Dr. Michael R. Powers, now Pro- fessor of Risk and Insurance Mathematics at Tsinghua University in Beijing, China, for his continued interest in the subject matter of this article and for fruitful discus- sions in this connection about the essence of probability in both its strictly mathematical and its more subjective interpretations. REFERENCES [1] L. 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 [2] Cf. J. Rosenhouse, “The Monty Hall Problem,” Oxford University Press, Oxford and New York, 2009 (history of the problem and its context). [3] Cf. R. Deaves, “The Monty Hall Problem: Beyond Closed vidence of the interest in the Doors,” 2006 (additional e problem). www.lulu.com [4] Cf. M. vos Savant, “Q(uestion) & A(nswer) (involving the Monty Hall Problem),” Parade, 9 September 1990 (article serving as principal catalyst of the interest in the problem). [5] H. H. Goldstine, “The Computer from Pascal to von Neumann,” Princeton University Press, Princeton, 1972, p. 37. [6] L. 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. 135-154, at p. 148. [7] A. W. F. Edwards, “Pascal’s Arithmetical Triangle: The Story of a Mathematical Idea,” Johns Hopkins University Press, Baltimore, 2002, p. xiii. [8] L. 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. 135-154, at p. 145. [9] L. Euler, “Elements of Algebra,” Springer Verlag, New York, Berlin, Heidelberg, and Tokyo, 1984, pp. 110-120. [10] S. F. Lacroix, “Traité élémentaire du calcul des prob- abilités,” Fourth Edition, Mallet-Bachelier, Paris, 1864, p. 30. [11] L. Depuydt, “The Monty Hall Problem and Beyond: Copyright © 2012 SciRes. APM L. DEPUYDT, R. D. GILL Copyright © 2012 SciRes. APM 273 [1 ed in what follows). [1 sa, Boole’s “×” (AND) is electrical ” Investigation of the Laws of Thought, on athematical Theories of Logic and Probabilities,” Walton and Maberly, London, 1854. [18] I have used thover Publications, ass., 1969, p. 214. ell, Vol. 3,” Routledge, London, 1993, pp. 366-379. . iv. ic 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, p. 150. [12] D. W. Miller, “The Last Challenge Problem: George Boole’s Theory of Probability.” http://zeteticgleanings.com/boole.html [13] G. Boole, “The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning,” Macmillan, Barclay, & Macmillan, Cambridge and George Bell, London, 1847. 4] G. Boole, “Studies in Logic and Probability,” Dover Pub- lications, Mineola, New York, 2004, pp. 45-124 (reprint of [13] whose pagination is us 5] G. Boole, “Studies in Logic and Probability,” Watts & Co., London, 1952 (original edition reprinted in [14]). [16] It should be noted that Boole’s “0” is electrical engineer- ing’s “1” and vice ver engineering’s “+” (AND) and vice versa, and Boole’s “+ (OR) is electrical engineering’s “×,” facts that I have failed to appreciate in the introduction to my “The Other Mathematics: Language and Logic in Egyptian and in General,” Gorgias Press, Piscataway, New Jersey, 2008, even if this oversight does not affect the arguments pre- sented in this work. It is difficult to find any published observations anywhere pointing explicitly to these fun- damental facts. Boole’s conventions are otherwise on oc- casion also used in engineering. [17] G. Boole, “An Which Are Founded the M and P e reprint of 1958 by D New York. [19] J. Venn, “Symbolic Logic,” Second Edition, Macmillan and Co., London and New York, 1894. [20] A. N. Whitehead, “A Treatise on Universal Algebra with Applications,” Cambridge, 1897, p. 11. [21] N. I. Styazhkin, “History of Mathematical Logic from Leibniz to Peano,” Cambridge, M [22] G. Boole, “Studies in Logic and Probability,” Dover Pub- lications, Mineola, New York, p. 53. [23] Th. Hailperin, “Boole’s Logic and Probability,” Second Edition, North-Holland Publishing Company, Amsterdam, New York, Oxford, and Tokyo, 1986. [24] B. Russell, “Recent Work on the Principles of Mathemat- ics,” International Monthly, Vol. 4, 1901, pp. 83-101, at p. 366 of the reprint in [25]. I owe the reference to [26]. [25] G. H. Moore (ed.), “The Collected Papers of Bertrand Russ [26] G. Bornet, “Frege’s psychologism criticism (of Boole),” In: I. Grattan-Guinness and G. Bornet, Eds., George Boole: Selected Manuscripts on Logic and Its Philosophy, Birkhäuser Verlag, Basel, Boston, and Berlin, 1997, pp. xlviii-l. [27] G. Boole, “The Mathematical Analysis of Logic,” Dover Publications, Mineola, New York, p. 47 G. Boole, “An Investigation of the Law[28] s of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities,” Walton and Maberly, London, 1854, p. 1. G. Bornet, “George [29] Boole: Selected Manuscripts on Logic and Its Philosophy,” In: I. Grattan-Guinness and G. Bornet, Eds., Science Networks Historical Studies, Birk- häuser Verlag, Basel, Boston, and Berlin, 1997, p. lx [30] G. Boole, “An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Log robabilities,” Walton and Maberly, London, 1854, p. 11. [31] G. Boole, “Studies in Logic and Probability,” Dover Pub- lications, Mineola, New York, p. 52. [32] G. Boole, “An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities,” Walton and Maberly, London, 1854, p. 11. |