Advances in Pure Mathematics, 2011, 1, 136-154
doi:10.4236/apm.2011.14027 Published Online July 2011 (http://www.SciRP.org/journal/apm)
The Monty Hall Problem and beyond:
Digital-Mathematical and Cognitive Analysis in Boole’s
Algebra, Including an Extension and Generalization
to Related Cases
Leo Depuydt
Department of Egy pt olo gy a n d Ancient W est ern Asi a n St u d i es , Brown University, Providence, Rhode Island, USA
Received April 27, 2011; revised May 14, 2011; accepted May 25, 2011
Abstract
The Monty Hall problem has received its fair share of attention in mathematics. Recently, an entire mono-
graph has been devoted to its history. There has been a multiplicity of approaches to the problem. These ap-
proaches are not necessarily mutually exclusive. The design of the present paper is to add one more approach
by analyzing the mathematical structure of the Monty Hall problem in digital terms. The structure of the
problem is described as much as possible in the tradition and the spirit—and as much as possible by means of
the algebraic conventions—of George Boole’s Investigation of the Laws of Thought (1854), the Magna
Charta of the digital age, and of John Venn’s Symbolic Logic (second edition, 1894), which is squarely based
on Boole’s Investigation and elucidates it in many ways. The focus is not only on the digital-mathematical
structure itself but also on its relation to the presumed digital nature of cognition as expressed in rational
thought and language. The digital approach is outlined in part 1. In part 2, the Monty Hall problem is ana-
lyzed digitally. To ensure the generality of the digital approach and demonstrate its reliability and productiv-
ity, the Monty Hall problem is extended in parts 3 and 4 to related cases in light of the axioms of probability
theory. In the full mapping of the mathematical structure of the Monty Hall problem and any extensions
thereof, a digital or non-quantitative skeleton is fleshed out by a quantitative component. The pertinent
mathematical Equations are developed and presented and illustrated by means of examples.
Keywords: Binary Structure, Boolean Algebra, Boolean Operators, Boole’s Algebra, Brain Science,
Cognition, Cognitive Science, Digital Mathematics, Electrical Engineering, Linguistics, Logic,
Monty Hall Problem, Neuroscience, Non-quantitative and Quantitative Mathematics, Probability
Theory, Rational Thought and Language
1. Introduction
The Monty Hall problem, named after the television host
Monty Hall who made it famous in a TV show, has re-
ceived its fair share of attention in mathematics. Recently,
accessibility to the history of the problem was greatly
enhanced due to the appearance of a monograph devoted
entirely to the subject [1]. A multiplicity of approaches
has been applied to the problem. These approaches are
not necessarily mutually exclusive. Among key contribu-
tions of most recent date to the problem’s analysis are an
updated statement of the Bayesian analysis of the prob-
lem [2], a challenge to move towards a mathematical
modeling of the problem [3], and yet other innovative
treatments [4].
The design of the presen t paper is to add one more ap-
proach by analyzing the mathematical structure of the
Monty Hall problem in digital terms. The structure of the
problem is described as much as possible in th e tradition
and the spirit, and by means of the algebraic conventions,
of George Boole’s Investigation of the Laws of Thought
(1854), the Magna Charta of the digital age, and of John
Venn’s Symbolic Logic (second edition, 1894), which is
squarely based on Boole’s Investigation and elucidates it
in many ways [5].
This paper has four main parts. The digital approach is

L. DEPUYDT
137
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outlined in general in part 1. In part 2, the Monty Hall
problem is analyzed digitally. The Monty Hall problem
involves 3 doors, 1 car, 2 goats, 1 picked door, and 1
opened door. To ensure the generality of the digital ap-
proach and to demonstrate its reliab ility an d produ ctiv ity,
it would seem to be critically important to extend and
generalize the analysis of the Monty Hall problem to any
number of doors, cars, opened doors, and picked doors in
light of the axioms of probability theory. Such an exten-
sion and generalization is the subject of parts 3 and 4 of
this paper. Part 3 concerns an extension to any number of
doors, cars, opened doors, and picked doors. Part 4 con-
cerns an additional extension to any number of picked
doors. The pertinent mathematical Equations are devel-
oped and presented and illustrated by means of exam-
ples.
2. Preliminary Considerations and
Reflections on Digitality and Cognition
2.1. Digital Mathematics and Quantitative
Mathematics
Digital mathematics is the mathematics in which nothing
gets bigger or smaller and everything is either On or Off,
1 or 0. Different notation systems are prevalent in dig ital
mathematics. I prefer the notation used by the Father of
the digital age, George Boole.
Digital mathematics needs to be differentiated from
the search for the roots of mathematics, a subject to
which Bertrand Russell and Kurt Gödel and many others
have contributed. These efforts are often called logistics,
as opposed to logic. A recent voluminous book on the
history of logistics from 1870 to 1940 documents all
these contributions in detail. This account also reveals
that it seems to be hardly the case that the ultimate roots
of mathematics have been once and for all fully uncov-
ered [5].
At the outset of his Elements of Algebra (Vollständige
Anleitung zur Algebra), Euler states that mathematics is
the science of quantity, the systematic study of that
which is capable of increase or diminution. This state-
ment is not fully complete. Digital mathematics is fun-
damentally different from the mathematics with which
one is better familiar.
In digital mathematics, nothing gets bigger or smaller.
When one performs a Boolean search on the Internet
looking for all that is Paris and in addition all that is
Paris–adding Paris to Paris as it were by using the
so-called Boolean OR-operator—the information that
one gets is not in any way larger than if one had just
searched for Paris alone. Adding Paris to Paris does not
produce a class or set that is twice as large as Paris. If p
is Paris, then


2pp p
in Boole’s algebra. By contrast,
in the familiar mathematics, quantity mathematics,

.
Furthermore, when one executes a Boolean search for
all entities that hav e two properties, namely b eing French
and again being French—multiplying French by French
as it were by using the so-called Boolean AND-opera-
tor—one does not obtain search information th at is larger
than if one had just searched for all that is French . If f is
French, then in Boole’s algebra,
f
ff
2
. By contrast,
in the more familiar mathematics, quantity mathematics,
f
ff. Something is getting bigger.

2.2. Boole’s Algebra and the Algebra of
Electrical Engineering
It should be noted that, when Boole’s algebra was
adopted in electrical engineering, the conventions were
switched. Boole’s “0” is electrical engineering’s “1” and
vice versa. Boole’s “×” is electrical engineering’s “+”
and vice versa [6]. I assume that Claude Shannon, who
adapted Boole’s algebra for the design of switching cir-
cuits in the 1930s and thus in a sense became the Father
of computer science, was the originator of this change. In
electrical engineering, 0 is conceived as zero resistance
or hindrance and therefore as an open circuit. I person-
ally prefer Boole’s notation and will use it in what fol-
lows. But as long as it is understood which functions
symbols have, it makes no difference whether one or the
other notation is used.
2.3. The Digital Nature of Rational Thought and
Language
The human experience consists entirely of how the brain
engages reality outside itself by means of the senses,
nothing more, nothing less. This includes any manner in
which the brain combines sensory perceptions internally.
There are more than the traditional five senses, sight,
hearing, smell, taste, and touch. The others include sens-
ing pain, sensing that one is upside down, sensing resis-
tance when pushing, and sensing hunger. Part of the
brain’s engagement with what is outside itself may be
called rational thought and language—as distinct from,
say, emotions.
I refrain from defining at this point exactly what is ra-
tional thought and what is language in rational thought
and language. It is much preferable to begin by regarding
the two together as a single large pheno menon. It may in
fact be difficult to disentangle the two entirely. After all,
to which of the two do any connections between the two
belong?
I am personally convinced that rational thought and

L. DEPUYDT
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1c
language is entirely digital. This fun damental assumptio n
will serve as a working hypothesis. I have begun con-
structing a comprehensive model of how rational thought
and language proceeds digitally. But the presentation of
this model is reserved for another occasion. In seeking
inspiration as to what this model might look like in the
brain, there is nothing wrong with creating physical tem-
plates consisting of magnetic coils and switches or tran-
sistors or memristors in an attempt to create a prefigura-
tion of what will be found later in th e brain.
The proposed model has nothing to do with any of the
many programs now in ex istence that allow a machine to
comprehend, produce, or translate languages. These pro-
grams are impressive, as appears from a translation by
Google Translate of a passage of Antoine de Saint-
Exupéry’s Le petit prince presented in a recent article in
the New York Times [7]. These programs do not, how-
ever, in any way mimic human language nor do they
pretend to. All are based on probability and statistics.
Relying on huge databases and much computing power,
the programs mathematically predict what is most likely
to come next based on information already stored. These
intelligent guesses are fast beco ming ever more accurate.
2.4. Empirical Basis for Observing the Digitality
of Rational Thought and Language
Little or nothing is known directly about how the bio-
logical brain produces thought and language. The ques-
tion arises: Is it not premature to construct models per-
taining to how the brain thinks and talks? Where is the
empirical basis?
The empirical basis is twofold. First, it is abundantly
clear that the brain teems with digital activity, even if
the precise mechanisms of this activity are mostly not
understood. Second, as one brain communicates with
another through thought and language, all communica-
tions need to travel by air from the mouth of a speaker to
the ear of a hearer or by light from the written page to a
reader’s eyes. There can be no doubt that ever ything that
is essential to the structure of rational thought and lan-
guage must be conveyed in sound waves or light beams
that travel from mouth to ear or from page to eye.
One might object that sounds and written symbols are
not the same as operations of neurons inside the brain.
Then again, certain operations of neurons generate a
structure that is empirically accessible in language. If
the structure of the neurons differed from the linguistic
structure they spawn, people would say things that differ
from what they think that they are saying. Clearly, the
structure expressed in language must be exactly the
same as the structure formed inside the brain, even if the
material platforms that the two inhabit could hardly dif-
fer more.
What about the validity of the proposed digital model?
It is true that mathematical models have predictive value.
Consider the computations relating to a novel kind of
bridge construction. The computations are predictive in
the sense that, if they are error free, the bridge should not
collapse. What is more, the computations are binding.
The bridge must be built according to the computations
or it will collapse.
The validity of digital mathematics in g eneral h as been
amply demonstrated by countless applications in tele-
phone circuits and computer science. Still, one cannot
conjure up just any fanciful digital analysis of brain op-
erations and simply expect the brain to operate according
to it. The digital analysis must meet certain empirical
conditions and be comprehensive in the mathematical
sense by extending to all possible cases. The digital
analysis should be to linguistic reality what mathematics
is to physical reality in the field of physics.
2.5. The Digital Supplements
When one looks at a page of written text, the 1s and the
0s do not readily jump at the eye. So where is the digital
structure? In a course that I might one day teach about
the digital nature of rational thought and language, I
might begin by confronting students with the expression
“two black cats” and ask where the mathematics is in this
expression. I would suspect that quite a few might point
to “two” as the mathematical component. However,
“black cats” is just as mathematical.
In digital terms, the presence of something creates a
certain awareness of its absence, in other words, of all
that it is not. Accordingly, to the class or set of cats cor-
responds a supplement class or supplement of all that is
not cats. If the class of cats is denoted by c, then its su p-
plement is denoted in Boole’s notation by

, that is,
the universe or all that one could possibly think about
(Boole’s “1”) minus (–) cats (c), or also by c. Likewise,
the class of all that is black can be denoted by b and its
supplement, all that is not black, by b
bc
.
2.6. The Digital Combination Classes
Furthermore, digitally speaking, two classes “black” (b)
and “cats” (c), along with their respective supplements,
divide the universe or all that one could possibly think
about into exactly four combination classes, black cats,
non-black cats, what is black but not a cat, and what is
neither black nor a cat. The present writer belongs to th e
fourth category. In Boole’s notation, the combination
classes are denoted by

(or bc), bc (or bc ),
bc (or

bc ), and bc (or

bc ). The universe or all

L. DEPUYDT
139
that one could possibly think about, Boole’s “1,” consists
entirely of the sum of the four combination classes, as
follows.
1
(or also:1
bcbc
bc

 )
bcbc
bcbc bc

 (1)
This division is profoundly digital. It encompasses all
the combinations in which presence and absence, or On
and Off, or 1 and 0, of two classes “cat” and “black” and
their supplements can combine.
The division is also fundamental to how we think.
Consider the simple sentence “The cat is black.” The two
classes “the cat” and what is black generate four combi-
nation classes. What matters is the abolition of one of the
four combination classes, namely what is both the cat
and not black. It is this operation of abolition that makes
the thought “The cat is black” possible.
In Boole’s algebra, there are two levels of thought, the
primary level of th e things and the secondary level of the
events. A proposition such as “The sun shines” is pri-
mary. The secondary level can be comprised of two pri-
mary propositions. An example is “When the sun shines,
I take a walk on the beach.” In this sentence, one digital
combination class is switched off or shut down or empty,
that is, all the occasions when the sun shines and I do not
take a walk on the beach. There is no such thing accord-
ing to said statement. The three other digital co mbination
classes are on or occupied: either the sun shines and I
walk on the beach, or the sun does not shine and I walk
on the beach, or the sun does not shine and I do not walk
on the beach.
The relation to the conditio sine qua non can be ex-
plained digitally [8]. Compare the statement already
mentioned, “When the sun shines, I take a walk on the
beach,” with the statement “Only when the sun shines do
I take a walk on the beach.” A different digital combina-
tion class is switched off in the latter statement, namely
all the occasions when the sun does not shine and I do
take a walk on the beach.
Furthermore, in the sentence “I take a walk on the
beach if and only if the sun shines,” both said combina-
tion classes are switched off. Since there are four digital
combination classes, that means that two combination
classes remain switched on: either I walk on the beach
and the sun shines or I do not walk on the beach and the
sun does not shine.
2.7. Limits to the Universe
When two classes and their supplements partition the
universe or all that one could possibly think about into
four combination classes, it is common to impose un-
spoken or explicit limits on what is being partitioned [9].
One hardly always considers everything thinkable. For
example, the statement “Manchester is the winner” de-
scribing the outcome of a soccer match between Man-
chester United and Liverpool involves four digital com-
bination classes: all that is Manchester and the winner,
all that is Manchester but not the winner, all that is not
Manchester yet the winner, and all that is neither Man-
chester nor the winner. The design of the statement
“Manchester is the winner” is to present two combina-
tion classes as empty: all that is Manchester but not the
winner and all that is not Manchester yet the winner. It
seems clear that all that is not Manchester does not in
this case include the Queen of England or the Pope in
Rome. Non-Manchester is limited to the soccer club
Liverpool. It is also clear that “Manchester” does not
refer to all of the city of Manchester, but just to the soc-
cer club Manchester United. Although the limits i mposed
on the universe are not stated explicitly, it seems clear
what they are.
2.8. Excursus: On Negation and on the Digitality
in Rational Thought and Language
In a digital world, negation is the mother of all meanings.
Everything without exception can be negated: “Caesar”
as “not Caesar,” “It rains” as “It does not rain,” “there”
as “not there,” “There is” as “There ain’t,” and “yes” as
“no.” It is as if reality presents itself to us in two parallel
universes, the affirmative and the negated. But what is
negation?
E. Schröder, the mathematician and onetime director
of the Technische Hochschule in Karlsruhe, advised
great caution when it comes to defining negation because
the most famous philosophers from Aristotle to Kant
proposed definitions of negation that are very far apart
and great authorities constructed untenable theories
about negation that exhibit the greatest internal contra-
dictions [10]. Greek philosophers struggled mightily with
being and not being and being and b ecoming and th e like.
But it is only since the mid-nineteenth century that digi-
tal mathematics has provided what I believe to be the
valid and final definition of negation. What is negation in
digital terms?
As the brain engages reality outside itself as observed
or as remembered or even as recombined, it naturally
does not focus on, or contemplate, everything all at once.
It selects certain components of what Boole calls the
universe, that is, all that one could possibly think about.
To a certain degree, by the way it is structured, reality
presumably more easily draws the attention of the brain
to certain of its facets rather than to others. One may

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L. DEPUYDT
focus on an entity such as a tree, on a property that
comes with that entity such as green, or on a circum-
stance in which that entity along with its property finds
itself such as “in the forest.” The contemplation of any
component of all that one could possibly think about
automatically conjures up the notion of all that remains,
all that that component is not, for example, all that is not
a tree, all that is not green, or all that is not in the forest.
If the component of the universe of thought is viewed
as a class or set, then all that is not that component is just
as much a class. A class that encompasses all that some-
thing is not may be called a supplement class or a sup-
plement. As was noted above, in Boole’s algebra, if
lower case g denotes all that is green, then all that is not
green is denoted by
g
, that is, the universe (Boole’s
“1”) minus g, which Boole abbreviates as
g
.
Negation is born, and so is digitality, when the need
arises to refer explicitly to all that something is not. For
example, I may want to state that a certain car falls out-
side the class of all that is green. I can do so by means of
the word “not,” as in “This car is not green.”
Importantly, a class and its supplement together make
up all that is thinkable. In this connection, Aristotle for-
mulated the fundamental axiom of thought, namely that
something cannot at the same time be and not be some-
thing. What is more, the relation between a class and its
supplement is like a toggle. The original class is of
course all that its supplement class is not. Put differently,
the original class is the supplement of its own supple-
ment. One common way of referring to this property of
digital reality is that two negations cancel one another.
This is not the place to illustrate the digital component
of rational thought and language at length. Suffice it to
point to a semantic field among whose members are
English words such as “alone,” “also,” “only,” “other,”
and “self” and an English expression such as “for his
part” along with their equivalents in other languages [11].
Everyone knows how to use these words. But defining
them is another matter. Without entering into detail, it
would appear that these words all refer to digital sup-
plement classes. For example, “he alone” means “no
others besides him,” “no non-he’s” as it were. “He also”
means “others besides him,” that is, “non-he’s” in addi-
tion to him. In a digital world, there is also a need for
referring with an explicit word to the supplement class.
The word “other” performs exactly that function.
“Other” refers to what something else that has been men-
tioned is not.
In this connection, I have also proposed to analyze
contrastive emphasis digitally [12]. When one says, for
example, “It is in Paris that the session is held” or “The
session is held in Paris,” one apparently means “not
somewhere else,” that is, “not in non-Paris.” Paris is
presented as the digital supplement of its digital supple-
ment, which is very much Paris itself.
3. Two Goats and a Car:
Digital-Mathematical Analysis of the
Monty Hall Problem
3.1. Boole and Probability
It is now generally forgotten that Boole wrote the Magna
Charta of the digital age, his Investigation of the Laws of
Thought (1854), to address problems in probability. But
his contribution to probability has been “simply bypassed
by the history of the subject” [13]. Boole believed that
the theory of probability is a field of mathematics strad-
dling the fence that separates quantitative mathematics
from digital mathematics.
It is assumed here as a working hypothesis that the
digital approach permeates all engagement of the brain
with reality, and that includes assessments of probability.
In support of this assumption, the Monty Hall pro blem is
analyzed digitally in the present part 2 and then extended
and generalized in ligh t of the axioms of probability the-
ory in parts 3 and 4.
The study of the Monty Hall problem has a long his-
tory. But the countless technical and popular treatments
of the problem are characterized by the exclusion of a
potentially fertile additional approach, the digital and
Boolean perspective, the perspective that I personally
believe reflects the fundamentally digital nature of how
the brain processes reality in terms of rational thought
and language.
3.2. Description of the Monty Hall Problem
Behind 3 closed doors, 2 goats and 1 car are hiding. One
is asked to pick a door to get what is behind it. The aim
is to get the car. One begins by picking a closed door
without however knowing or being told what it is hiding .
Subsequently, someone who knows what is behind all 3
doors without revealing this knowledge to the person
who has picked a closed door opens 1 of the 2 doors that
were not picked, and more specifically a door hiding a
goat. Since 2 of the 3 doors hide goats, it is always pos-
sible to open a non-picked door that hides a goat. The
other 2 doors remain closed and 1 of these 2 is the one
that was initially picked.
The Monty Hall problem revolves entirely around the
following question. Once a door has been opened to re-
veal a goat, should one switch from the closed door that
one has picked to the other door that remains closed in

L. DEPUYDT
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order to improve one’s chances of getting the car? There
is no doubt whatsoever that one should switch. The need
is therefore for establishing the respective probabilities
of switching and not switching doors.
3.3. Intuitive Grasp of the Need to Switch Doors:
Opening Doors to Reveal Goats as a Means
of Compressing or Condensing a Probability
into Fewer Doors
The need for switching doors to improve one’s chances
of getting the car is obvious from the following consid-
eration. Everyone knows that one has only 1 chance in 3
of getting the car by picking a certain door. That in effect
means that there are 2 chances in 3 that the car is hiding
behind 1 of the 2 other doors that were not picked. In
other words, there are 2 chances in 3 that the 2
non-picked doors hide 1 car and 1 goat. Accordingly,
there ought to be every temptation to switch to the other
2 doors to get the car. The problem is that one cannot
switch to both of the 2 other doors that were initially not
picked. One can only switch to 1 of them. But to which
one of the 2 should one switch?
This is where critical and odds-changing help arrives
in the form of someone opening 1 door to reveal 1 goat.
The effect of this intervention is that the chance of 2 in 3
that the 2 other doors are hiding the car is concentrated in
1 unopened door. What opening 1 door to reveal 1 goat
achieves is to compress or condense a certain degree of
probability distributed equally over a number of doors
into fewer doors. In the case of the Monty Hall p roblem,
a probability of 23, which is distributed equally as a
probability of 13
i
C
over each of the 2 doors that are ini-
tially not picked, is compressed into 1 door by opening 1
door to reveal 1 goat.
In sum, there is every reason to switch doors. It is not
certain that one will get the car. Bu t one has 2 chan ces in
3 of being lucky.
3.4. The Two Digital Levels, the Level of Things
and the Level of Events
Just as two classes “black” and “cat” digitally generate
four combination classes (§2.6), the Monty Hall problem
in a digital and Boolean perspective fundamentally in-
volves two classes generating four combination classes.
But in the case of the Monty Hall problem, the two
classes do not contain all the instances of two thin gs,
such as “black things” and “cats,” but rather all occa-
sions or instances or occurrences of two events. Accord-
ing to Boole’s analysis, rational thought and language
exhibits two levels, the level of primary proposition s that
is concerned with classes of things and the level of sec-
ondary propositions that is concerned with classes of
events. Each class of events contains all the occurrences
of a certain event.
When two classes of events and their supplement
classes are partitioned into four combination classes, it is
common to impo se un spok en o r explicit li mits o n what is
being partitioned (§2.7). The same applies to the Monty
Hall problem and any extensions thereof. What one is
considering in terms of things is limited to a number of
doors hiding either cars or goats. What one is consider-
ing in terms of events is limited to initially pickin g or not
picking a car and then picking or not picking a car by
switching after one or more doors have been opened to
reveal goats.
3.5. Mathematical Notation of Things and
Events
To differentiate things and events in mathematical nota-
tion, things are denoted below by lower case italic letters
and events by upper case italics. For example, c stands
for “cars” and C for picking a door hiding a car. In
addition, subscript letters provide additional distinctive
information about events. For example, will stand for
initially picking a door hiding a car.
3.6. The Two (Classes of) Events Involved in the
Monty Hall Problem, the Second Dependent
on the First
In the case of the Monty Hall Problem, two events are
involved. What is more, the two events occur in a fixed
sequence. The second event always follows the first and
is dependent upon the first according to the definition of
dependence in the classic theory of probability.
The two classes of events are as follows. The first
contains all the occasions when one initially picks the
door behind which the car is hiding. The second contains
all the occasions when one picks the door behind which
the car is hiding by switching from the door initially
picked to the sole door that remains closed after a door
has been opened to reveal a goat.
3.7. The Supplement Classes, or Supplements, of
the Two Events Involved
Digitally speaking, all classes come with supplement
classes or supplements. A supplement contains all that a
class is not. A supplement is a class in its own right. On
the level of things, the supplement of “cat” is all that is
not a cat. On the level of events, the supplement of “It
rains” is all the occasions when it does not rain.
In the case of the Monty Hall problem, the supplement

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of initially picking the door hiding the car contains all
the occasions when one fails to initially pick that door.
The supplement of picking the door hiding the car by
switching from the door initially picked to the sole other
door remaining closed after a door has been opened to
reveal a goat contains all the occasions when one fails to
pick the door hiding the car by switching doors.
3.8. The Two Classes and Their Supplements in
Boole’s Algebra
In Boole’s algebra, picking the door hiding the car may
be denoted by and failing to pick that door byC
1
i
C
.
Boole uses overstrike as an abbreviation of 1–, as in
, that is, the universe or any events that one could
possibly think about minus (–) all the occasions when the
door hiding the car is picked.
C
Initially (i) picking the door hiding the car may be de-
noted by and failing to initially pick that door by
i
C. Furthermore, picking the door hiding the car by
switching (s) doors once a door has been opened to re-
veal a goat may be denoted by
s
Cand failing to pick that
door by switching doors by
s
C
i
C
.
Since initially picking a car () is the same as ini-
tially not picking a goat (i G), and so on, the following
Equations apply.
ii
CG ii CG
s
s
CG
s
s
CG
CC
3.9. The Four Digital Combination (Classes of)
Events
In digital fashion, two classes of events along with their
supplements generate four combination classes corre-
sponding to all the four possible combinations of the
occurrences and non-occurrences of the two events. The
four combination events are as follows:
(1) (On-On) initially pick the car and then again pick
the car by switching doors;
(2) (On-Off) initially pick the car but then fail to pick
the car by switching doors;
(3) (Off-On) initially fail to pick the car but then suc-
ceed in picking the car by switching doors; and
(4) (Off-Off) initially fail to pick the car and then
again fail to pick the car by switching doors.
Digital combination class (1) is denoted in Boole’s
notation by is or as is CC ;is is yet another
notation. Boole’s “×”—now better known as the Boolean
AND-operator even if Boole himself did not refer to it in
this—manner denotes a combined event in which two
component events are valid at the same time. The other
three combination events may be denoted by
CC
s
i
CC , is CC ,
and
s
i
CC
C
.
3.10. The Probabilities of the Digital
Combination Events
In Boole’s algebra, four symbols such as i, i, C
s
C,
and
s
C
CC
not only denote classes of events but also the
probability that the events in question will take place.
Accordingly, the probabilities of the four digital combi-
nation events (1), (2), (3), and (4) listed in §3.9 consist
for each combination event of the product of the prob-
ability that the first of the two combined events will take
place multiplied by the probability that the second event
will. The four probabilities may therefore be denoted by
is

,
s
i
CC

,is CC

, and
s
i respectively.
The symbol “×” is then to be understood quantitatively
and not as an equivalent of what is now known as the
Boolean AN D- operator.
CC
C
3.11. 100% as the Sum of the Probabilities of All
Combination Events
The two classes “cat” and “anything black” along with
their supplements subdivide all that is thinkable in terms
of things, as denoted by Equation (1) in §2.6. Likewise,
the two classes i and
s
C, along with their supple-
ments subdivide all that is thinkable in terms of events,
as denoted by Equat i on (2).
1isisis CC CC CC


is
CC
C
(2)
In other words, the totality of all possible scenarios
consists entirely of four combination events: either i
and
s
C C both happen, or i does and
s
C does not, or
i does not but C
s
C does, or neither i nor C
s
C
i
C do. Put differently, either and
s
C co-occur, or i C and
C
s
do, or i and
C
s
C do, or and
i
C
s
C do. There
are no other possibilities.
Accordingly, the probabilities of the four combination
events add up to 1 or 100%. It is one hundred percent
certain that one of the four combination events will take
place.
3.12. The Two Empty Digital Combination
Classes of Events
Closer reflection reveals that two of the four combination
events involved in the Monty Hall problem never occur,
namely is CC

and
s
i
CC

. The corresponding classes
of events are therefore empty. Digitally speaking, these
two combination events are switched off, as it were.

L. DEPUYDT
143
CC
As regards is
, one cannot initially pick the car
and then again pick the car by switching doors because
there is only 1 car. As regards
s
i, one cannot ini-
tially fail to pick the car—in other words, pick a goat—
and then again fail to pick the car—that is, again pick a
CC
goat—by switching doors. The reason is that, when one
has initially picked a goat, the door hiding the only other
goat is opened and one cannot switch to that door be-
cause it is now open. Accordingly, the only door to
which one can switch hides the car and one cannot fail to
pick the car by switching doors.
The probability of these two combination events is
therefore equal to zero (0). Equation (2) in §3.11, which
denotes the sum of the probabilities of the four digital
combination events, can therefore be reduced to a sum of
two combination events, as follows.
i
s
is
CC CC1 (3)
3.13. The Digital Combination Event Including
the Desired Outcome of Getting the Car
(C), namely 
is
CC
C
The desired outcome of the Monty Hall problem as a
challenge is getting the car (). Of the two digital com-
bination classes of events that are not empty in Equation
(2) in §3.12, namely is
and CCis
CC, only is CC

includes the desired outcome C. Importantly, this com-
bination event involves switching doors, as denoted by
subscript s, after initially failing to pick the car (i). In
the other combination event, one had initially picked the
car , but then loses it by switching doors.
C
Ci
If the probability of the combination event with the
desired outcome is lower than 0.5 or 50%, one should
not switch doors to increase one’s chances of getting the
car. If the probability is higher than 0.5 or 50%, switch-
ing doors makes getting the car more probable. If the
probability is exactly 0.5 or 50%, it does not make a dif-
ference whether one switches doors or not; one does not
increase or decrease one’s chances of getting the car.
What is the probability of the combination event that
includes the desired outcome?
Two of the four digital combination classes of events
are empty (§3.12). Accordingly, the sum of the prob-
abilities of the two other combination events is 1 or
100%. Either one or the other of the two other combina-
tion events must be the case.
If the probability of one of the two combination events
is x%, then the probability of the other is (1
3.14. The Probability of the Digital Combination
Event Including the Desired Outcome,
namely
x
)% be-
cause there are only two. The probability of one combi-
nation event can be derived from the probability of the
other by subtracting the probability of the other from 1 or
100%.
is
CC

The desired outcome is achieved by switching doors as
part of the digital combination event is . In order to
compute the probability of this combination event, it is
first necessary to establish the probabilities of its two
components, the two events
CC
i
C and
s
C.
The probability of i C, that is, failing to initially pick
the car, is evidently 23 or about 66.7%. There are 3
doors and 2 of them hide a goat. C
In establishing the probability of
s
, it needs to be
taken into account that
s
C is a dependent event ac-
cording to the definition of dependence in classical
probability theory. Accordingly, the question to ask is:
What is the probability that one will pick the car by
switching (
s
C) after having initially failed to pick it (i
C)?
That probability is 1 or 100%. Indeed, if one first picked
a goat and the other door hiding a goat is opened,
switching to the only other unopened door must always
result in picking the car.
The probability of the desired combination event i
× C
s
C, which results in picking the car by switching
doors (
s
C), is therefore 23 123 . Not only does this
combination event produce the desired outcome but its
probability is also higher than 0.5 or 50%. There can be no
doubt: One has to switch doors to improve one’s chances
of getting the car.
3.15. The Probability of the Combination Event
That Fails to Achieve the Desired Outcome,
namely s
i
CC

Failing to get the car is the outcome of the digital com-
bination event i C
s
C

. After initially picking the car
(i), one loses it by switching doors (C C
s
). The prob-
ability of this digital combination event can be derived
directly from the probability of the other digital combi-
nation event is CC

(§3.14). That is because only two
of four possible digital combination events actually oc-
cur in the Monty Hall problem. Their combined prob-
ability must therefore be 1 or 100%. Either one or the
other must take place. Since the probability of is CC

is 23 (§3.14), the probability of
s
i
C must be C
123 or

13. But for completeness’ sake, it may be
desirable to establish the probability of
s
i
CC

in its
own right.
The probability of initially picking the car (i) is C13
or about 33.3%. There is 1 car and there are 3 door s. But
once the car is chosen, the probability of failing to pick
the car by switching doors ( C
s
) is 1 or 100% because
one has already picked the sole car and cannot pick it

144 L. DEPUYDT
again. Consequently, the probability of
s
i
CC is 131

or 13

.
4. Extension and Generalization of the
Monty Hall Problem to Any Number of
Doors (d), Cars (c), and Opened Doors (o)
4.1. The Probability of Getting a Car by
Switching Doors (Cs):


cd-
- -o
1
1 dd
The general question that stands at the center of the lar-
ger problem of which the Monty Hall problem represents
just one specific case is as follows: Should one switch
doors in order to improve one’s chances of getting cars
after doors have been opened to reveal goats? To answer
the question, one needs to know the respective probabili-
ties of picking a door hiding a car by switching doors
(
s
C) and picking a door hiding a car by not switching.
The probability of picking a door hiding a car by not
switching is the same as the probability of initially pick-
ing a door hiding a car (i C). If one does nothing else, the
probability does not change. If the probability of getting
a car by switching is greater than the probability of get-
ting a car by not switching, then one should switch doors
to improve one’s chances of getting a car. A comparison
between the probabilities of
s
Cand imposes itself.
i
Of the two probabilities in question, that of picking a
door hiding a car by not switching is readily known. As
was said, it is the same as the probability of initially
picking a door hiding a car (i C). This probability is the
ratio of cars to doors and can be denoted by
C
cd. It is
the ratio of favorable outcomes to the sum of favorable
and unfavorable outcomes, in accordance with a funda-
mental axiom of classical probability theory. For exam-
ple, if there is 1 car and there are 3 doors, the chance of
initially picking a door hiding a car is 13 and 13
remains the chance of getting the car if one does nothing
else, such as switching to another door. If there are 2 cars
and 5 doors, the chance of getting a car by sticking with
the door that was initially picked is 25. And so on.
As regards the other probability, that of getting a car
by switching doors (
s
C), Equation (4) applies.



1
1
cd
dd o

s
C

 (4)
c = the number of cars (=
g
, the number of non-goats)
d = the number of doors (=, the sum of cars and
goats) cg
1p1p
dcg cdg
o = the number of opened doors
How Equation (4) is obtained is explained below in
§§4.4-6. In Equation (4), 1 is the number of picked doors
(p). The picked doors are not denoted algebraically by,
say, p because Equation (4) is only valid when there is
only 1 picked door, that is, when . Cases in which
there is more than 1 picked door, that is, in which ,
are discussed in part 5 bel o w.
Equation (4) does not feature goats (g). However, of
the three variables d, c, and g, each can be derived from
the two others because , , and
g
dc

.

When the number of the cars is 1, Equation (4) can be
simplified as follows.

scg g
Cdg odg o 

(5)
After all, (4) is equivalent to the following Equation .



1
1
s
cc g
Cdc go

 (6)


1c If

, then the following Equation can be substi-
tuted for (6).


11 1
11
s
g
Cdg o

 (7)
And Equation (7) can be rewritten as Equation (5). As
will be seen below, the deeper reason for this simplifica-
tion is as follows. When there is more than 1 car, getting
a car by switching doors can be the result of both initially
picking a door hiding a car and initially picking a door
hiding a goat. There are two scenarios. When there is
only 1 car, getting the car by switching doors can only be
the result of initially picking a door hiding a goat. There
is only one scenario. One cannot initially pick the only
car and then pick it again when switching doors.




d-d- -o11
4.2. as the Factor by Which
the Probability of Getting a Car by Not
Switching Doors Increases by Switching
Doors After Doors Have Been Opened to
Reveal Goats
Equation (4) can also be rewritten as follows.
1
1
scd
Cddo

 (8)
The first member of the right-hand term in Equation
(8), cd, is the probability of getting a car by not
switching. As was noted earlier, this probability is the
same as the probability of initially picking a door hiding

L. DEPUYDT
145
Ca car (i). The probability i C cannot change if one
does nothing else. Equation (8) can therefore also be
written as follows.
1
1
sid
CC
do



 
(9)
The second member of the right-hand term in Equa-
tion (9),

1dd1o, is the factor by which the
probability of getting a car by not switching doors is
changed by actually switching.
When there is only 1 car, then Equation (8) can be
written in light of Equation (5) in §4.1 as follows.
scg
dgo


C (10)
The factor by which the probability of getting a car by
not switching doors is changed by actually switching is
then

g
go0o .
If no doors are opened to reveal goats, then

and the factor in question is




1d1d (or
g
g
1
when there is only 1 car), that is, 1. In this specific case,
Equation (9) is equiv alent to the following Equation.
s
ii
CC C
0
o

(11)
The probability does not change by switching doors
and one hence does not increase o ne’s chances of gettin g
the car by switching.
If one or more doors are opened, then o and
is smaller than . It follows that 1d1d

1dd1o is larger than 1 when any doors are
opened. In this case, Equation (9) is equivalent to the
following Equatio n.
s
i
CC (12)
In sum, one will always improve one’s chances of get-
ting a car if one switches doors after doors have been
opened to reveal goats. The increase in the probab ility of
getting the car will vary. For example, it will be small if
there are many, many doors and only a couple of doors
hiding goats are opened. But there will always be an in-
crease.
What is more, any increase in the number of opened
doors (o) produces an increase in the probability of get-
ting the car by switching doors (
s
C

). As o in

1dd1do 1o increases, decreases. Ac-
cordingly, the fraction as a whole increases and so does
s
C.
Any increase in the number of doors hiding cars (c)
increases the chances of getting a car in genera l and any
increase in the number of doors hiding goats (g) de-
creases those chances in general. But any changes in the
number of cars or doors do not affect the probability of
getting a car by switching doors. Only the number of
opened do ors (o) does.
4.3. Examples
4.3.1. Openin g Do o rs t o Reveal Goats Always
Increases the Probability of Getting a Car by
Switching Doors
Equation (4) in §4.1 defines the probability that one will
get a car by switching doors and makes it possible to
compare this probability with the pro bability of getting a
car by not switching. It is a fact that the probability of
getting a car will always remain th e same when no doors
are opened to reveal goats and will always in crease when
one or more doors are opened. There is therefore never
any point in switching doors to get a car when no doors
are opened to reveal goats and always reason to switch
when do ors are opened.
The factor by which the probability in question in-
creases by switching doors when doors are opened is




11ddo



in general and
g
go in the
special case in which there is only 1 car (§4.2).
4.3.2. The Special Status of the Monty Hall Problem
The Monty Hall problem is just one instance of a much
more general problem. The Monty Hall problem is spe-
cial in that it presents what may be called the minimal
scenario of the general problem. It exhibits the lowest
numbers of doors, cars, and opened doors. The number
of cars cannot be lowered because there is only one. The
number of door s cannot be low ered fro m 3 to 2—and the
number of goats therefore from 2 to 1—because it then
becomes impossible to always open a door revealing a
goat. By using Equation (4), it can be determined that the
chances of getting the car by switching are as follows.





1131
2
133113
cd
dd o




Moreover, the probability of getting the car is in-
creased through switching by a factor of 1
1
d
do


, in
this case of



13 12
3311




.
Because there is only 1 car, it is possible to use Equa-
tion (5) in §4.1. The probability of getting the car by
switching doors is then determined as follows.

22
32 13
g
dg o




146 L. DEPUYDT
In any variations on the Monty Hall problem, the
numbers of do ors hiding cars and goats and opene d do or s
can only be increased. Increasing the doors hiding cars
increases the probability of initially picking a car and
therefore also of getting a car by not switching. Increas-
ing doors hiding goats decreases that same probability.
But neither will increase the probability of getting a car
by switching as op posed to not switching. Th at probabil-
ity is increased only b y increasing the number of opened
doors.
4.3.3. Chan ging the Number of Doors ( d) a nd Doors
Opened to Reveal Goats (o)
The most characteristic effect of the Monty Hall p roblem
and its variations is the way in which opening doors to
reveal goats increases the probability of getting a car by
switching doors. What is more, the increase of the prob-
ability of getting a car by switching doors grows as ever
more and more door s are opened to reveal goats. But th is
increase can only grow on condition that there are more
doors hiding goats in the first place.
For example, the number of doors hiding goats could
be increased to 9 and the total number of doors therefore
to 10. If only 1 door is opened to reveal a goat, the prob-
ability of getting the car by switching doors (
s
C) is, in
light of Equation (4) in §4.1, as follows, given 1 car (c),
10 doors (), a n d 1 opened door (). do




11
11
cd
dd o
101
9
0101180


 or 11.25%
Because there is only 1 car, it is possible to use Equa-
tion (5) in §4.1. The probability of getting the car by
switching doors is then determined as follows.

99
9 180

10
g
dg o
The probability of getting the car by not switching is
110 or 10%. The increase in the probability of getting
the car by switching is only 1.25%. The difference be-
tween switching and not switching would only become
noticeable as the process of picking a door by switching
doors is repeated over and over agai n. But as the opened
doors (o) increase in number, so does the probability of
getting the car by switching doors. If there are 2 doors,
the probability is as follows.

99
9 270

10
g
dg o or about 12.6%
As 6 more doors hiding goats are opened one by one
until only 1 unopened door hiding either a car or a goat
remains, the probability of getting the car by switching
doors gradually increases as follows: from 960 (15%)
to 950 (18%) to 940 (22.5%) to 930 (30%) to
920 (45%) and finally to 910

(90%). When only 1
door is left unopened besides the initially picked door,
the chances of getting the car are 9 times greater when
switching doors than when not switching doors and one
has a 90% chance if one switches. What is more, the
probability of getting the car by switching doors no less
than doubles when, of 2 doors remaining unopened be-
sides the picked door, 1 is opened and only 1 door is left
to which one can switch .
If there ar e 1000 doors of which 1 hides a car an d 999
hide a goat and 998 doors are opened to reveal goats, the
chance of getting the car by switching to the 1 door re-
maining unopened is as follows.

999 999
1000 9999981000
g
dg o 
One has a 99.9% chance of getting the car by switching
doors.
4.3.4. Chan ging the Number of Cars (c)
The examples provided have so far illustrated changing
the number of doors hiding goats (g) and changing the
number of opened doors (o). The main ai m of the exam-
ples was to illustrate the effect of opening doors in terms
of probability. Towards that end, the number of doors
being opened to reveal goats had to be increased. But this
number cannot be increased without increasing the
number of doors hiding goats. That still leaves changing
the number of cars to be illustrated.
As was noted above (§4.1 end), when there is 1 car,
there is one scenario, and when there is more than 1 car,
there are two scenarios. As will be seen below, when
there are two scenarios, the probability of getting a car
by switching doors consists of a sum of two terms. When
there is only one scenario, there is only one term.
Let us assume that c = 6, g = 4, d = 10. The probability
of initially picking a door hidin g car and th erefo re also of
getting a car by not switching doors is 610 or 60%.
The probability of getting a car by switching doors in-
creases as follows when first 1, then 2, and finally 3
doors are opened to reveal a goat.



opening 1 d oo r: 610 154
10 101180

 or 67.5%



opening 2 doo rs: 610 154
10 101270

 or about 71.1%



opening 3 d oors: 610 154
10 101360

 or 90%

L. DEPUYDT
147
i
i
4.4. Probability Theory as a Mathematical
Discipline That Combines the Digital or
Non-quantitative with the Quantitative
The design of §§4.4-7 is to clarify how Equation (4)
(§4.1) has been obtained. This Equation makes it possi-
ble not only to determine the probability of getting a car
by switching doors under the precise conditions stipu-
lated in the Monty Hall problem (see above) but also to
extend and generalize the Monty Hall problem to any
number of doors, cars, and doors that are opened to re-
veal a goat. However, getting a car is only one possible
outcome. The other outcome is not getting the car. Equa-
tion (4) is therefore only one component of a larger
mathematical problem. To ensure an adequate analysis
and a true appreciation of the Monty Hall problem itself,
it will be necessary to map the mathematical structure of
the larger problem. The larger problem concerns all pos-
sible outcomes . The total p robability of all outco mes is 1
or 100%. One or the other outcome of all possible out-
comes must always be the case.
But first, to ensure an even deeper appreciation of the
Monty Hall problem on an even more general level, it
will be useful to highlight a most peculiar general prop-
erty of the type of mathematics needed to analyze the
problem and its extensions, namely probability theory. In
that regard, no one more than Boole has taken pains to
emphasize—and in fact hardly anyone has in addition to
him—that prob ability theory is a mathematical discipline
straddling the digital or non-quantitative and the quanti-
tative.
The mathematical anatomy of the larger problem of
which the Monty Hall is a specific case hence has two
components, a digital or non-quantitative component and
a quantitative component. The relation between the digi-
tal component and the quantitative component is such
that the digital component provides a skeleton that is
fleshed out by the quantitative component.
4.5. The Digital or Non-quantitative
Component in the Mathematical Mapping
of the Monty Hall Problem and Its
Extensions
The larger problem of which the Monty Hall problem
presents just one specific case revolves entirely around
the four possible combination events in which either a
first event or its supplement or negation can be followed
by either a second event or its supplement or negation. In
the specific case of the Monty Hall problem, the prob-
ability of two of the four possible combination events is
zero.
Two classes, whether of things or of events, and their
supplements produce exactly four combination classes
(§2.6). Three events and their supplements would pro-
duce exactly eight combination classes, and so on. This
structure is digital or non-quantitative. One might rightly
object t h at at l ea s t t h e n umb er o f th e co mb i nat i o n clas s es
is quantitative. Strictly speaking, therefore, it is only the
exploitation of the contrast between a class and its sup-
plement and the combination of classes and their sup-
plements into combination classes that is digital.
Certain combinations of two classes and their supple-
ments are invalid. Combining a class with itself produces
just that class (§2.1) and is therefore not a valid combi-
nation. Combining a class with its supplement is not pos-
sible because, according to the fundamental axiom of
thought, something cannot at the same time be and not be
something (§2.8).
In the case of the Monty Hall problem and any exten-
sions thereof, the first event is initially () picking a door
hiding a car (C), denoted here by i C. The first event
can also be defined as failing to pick a door hiding a goat
or a non-car. The supplement or negation of the first
event is initially () failing to pick a door hiding a car
(C), denoted here by i C. The supplement may also be
described as picking a door hiding a non-car. The second
event is picking a door hiding a car () by switching (s)
doors and is denoted here by C
s
C. The supplement or
negation of the second event is failing to pick a door
hiding a car ( C) by switching (s) doors and is denoted
here by
s
C
is
CC
. In sum, there are exactly four digital com-
bination classes (of events), the following.

(1)
(2) is CC

(3) is
CC

(4) is CC

The total probability of the occurrence of these four
digital combination events is 1 or 100%. That means that
one of these four combination events must always be the
case. The four digital combination classes are also mutu-
ally exclusive. Combining them would result in combi-
nations of a class and its supplement. As has already
been noted, according to the fundamental axiom of
thought, such combinations are impossible. Something
cannot at the same time be and not be something.
Of the four digital combination classes, two have the
desired outcome of picking a car by switching doors (C
s
),
namely (1) and (3). It is on these two combination classes
that the study of the Monty Hall problem and any exten-
sions thereof focuses (see §4.7 below). But the problem
can only be mathematically appreciated to its full extent
by considering all possible outcomes, whose combined
probability is 1 or 100%.

148

L. DEPUYDT
4.6. The Quantitative Component in the
Mathematical Mapping of the Monty Hall
Problem and Its Extensions
4.6.1. Preamble: Boole’s “×” as an Expression of How
the Digital and the Quantitative Complement
One Another as Two Facets of a Single
Phenomenon
At the center of the mathematical structure of the larger
problem of which the Monty Hall problem presents just
one specific case stand four mutually exclusive digital
combination classes of events. In Boole’s algebra, they
are denoted by
is
CC , is
CC, i CC
s

, and is CC
CC
.
As a notation of digital combination classes, the sym-
bol “×” represents the Boolean AND-operator, which is
used for example when one searches for all that is both
Paris and hotels on the Internet.
However, in Boole’s algebra, the notations is

,
is
,
CCis CC, and is
can also stand for the
numerical probabilities of the occurrence of the four
combination classes. The symbol “×” then functions as
the multiplication sign of quantitative mathematics.
Boole’s notation system therefore most felicitously repre-
sents probability as a single phenomenon that exhibits
digital or non-quantitative properties and quantitative
properties inextricably linked to one another.
CC
The quantitative probability of a combination class of
events involving two events is obtained by first calculat-
ing the probability of the first event, then calculating the
probability of the second event, and finally multiplying
the two probabilities with one another.
In calculating the probability of the second event, it
needs to be taken in account that the second event may
be dependent on the first event according to the defini-
tion of dependence in classical probability theory. The
dependence is such that the occurrence of the first event
creates a new situation that influences the calculation of
the probability of the second event. In other words, the
second event takes place in a environment different from
the one in which the first event takes place. And it is the
first event that changed the environment.
4.6.2. The Probability of Digital Combination Event
1
1
cc
o


CC
C
dd
：
is
CC
To establish the probability of is , the probability of
the first event i initially (i) pickin g a door hiding a car
(C), needs to be determined first. This probability is the
ratio of all the cars to all the doors, which can be denoted
by cd. If there is 1 car and there are 3 doors, the
chance of picking the doo r hid i ng the car is 1 in 3.
Next is establishing the probability of the second event
s
C
1c
1d
, picking a door hiding a car (C) by switching doors
(s). This probability is also a ratio of cars to doors. Since
a car has already been picked in the first event, the num-
ber of cars needs to be reduced by 1 to . And since
one cannot pick again the door picked in the first event
because one needs to switch, the number of doors needs
to be reduced by 1 to

. As was already noted above,
the number of picked doors, 1, is not denoted algebrai-
cally by , say , p because the form ula be i ng deve lo p ed here
is only valid when 1 door is picked, that is, when 1p

.
Furthermore, since one cannot pick any doors being
opened to reveal a goat, needs to be reduced fur-
th er by t he num ber of op en ed doo r s (o) to . The
1d1do
probability of
s
C will therefore be 1
1
c
do


is
CC
as a ratio
of cars to doors.
The combined probability of will be
1
1
cc
ddo



. In the case of the Monty Hall problem, the
probability in question is as follows.
111 101
00
3311 313




This probability is zero. It is not possible to pick the
door hiding the car by switching doors if one has already
initially picked the car because there is only 1 car.
4.6.3. The Probability of Digital Combination Event

is
CC

:

1
cgo
ddo

As regards the probability of is , the probability of
the first event C has already been determined to be
CC
i
cd(§4.6.2).
Next is establishing the probability o f the seco nd ev ent
s
C, picking a door hiding a goat or non-car (C) by
switching doors ( s). This probability is a ratio of goats to
doors. Since no goat has been picked in the first event,
the number of goats does not need to be reduced in this
respect. However, sinc e one cannot pick any doors being
opened to reveal a goat, the goats in those doors cannot
be picked either and the number of the goats (g) needs to
be reduced by the number of opened doors (o) to
g
o

.
Furthermore, again because one cannot pick any doors
being opened to reveal a goat, the number of doors
available for being picked () needs to be reduced as
well by the number of opened doors (o), namely to
1d
1do

. The probability of
s
C will therefore be
g
1
o
do 
 as a ratio of goats to doors.
The combined probability of is CC will be

L. DEPUYDT
149
1
cgo
ddo

 . In the case of the Monty Hall problem, the
probability in question is as follo ws.
121
3311 3



111 1
1
1 33

In one third of all possible cases, one will initially pick
a door hiding a car and then pick a door hiding a goat by
switching doors.
4.6.4. The Probability of Digital Combination Event
 1
gc
dd o
 ：
is
CC
To establish the probability o f is , the probability of
the first event CC
i
C, initially (i) picking a door hiding a
goat or non-car (C), needs to be determined first. This
probability is a ratio of all the goats to all the doors,
which can be denoted by
g
d. If there are 2 goats and
there are 3 doors, the chance of picking a door hiding a
goat is 2 in 3.
Next is establishing the probability o f the seco nd ev ent
s
C
1d1d
1do
, picking a door hiding a car (c) by switching doors
(s). This probability is a ratio of cars to doors. Since no
car has been picked in the first event, the number of cars
(c) does not need to be reduced. But since one cannot
pick again the door picked in the first event because one
needs to switch, the number of doors needs to be reduced
by 1 to . Furthermore, since one cannot pick any
doors being opened to reveal a goat, needs to be
reduced further by the number of opened doors (o) to
. The probability of
s
C will therefore be
1
c
do as a ratio of cars to doors.
The combined probability of is CC will be
1
gc
dd o
. In the case of the Monty Hall prob lem, the
probability in question is as follo ws.
21
3311 3


212 2
1
1 33

In two thirds of all possible cases, one will initially
pick a door hiding a goat and then pick a door hiding the
car by switching do o rs.
4.6.5. The Probability of Digital Combination Event


1
1
gg o
dd o
 ：
is
CC
As regards the probability of is , the probability of
the first event
Next is establishing the probability o f the seco nd even t
s
C, picking a door hiding a goat or non-car (C) by
switching doors ( s). This probability is a ratio of goats to
doors. Since a goat has been picked in the first event, the
number of goats needs to be reduced by 1 to
g
1

. Fur-
thermore, since one cannot pick any doors opened to
reveal a goat, the goats in those doors cannot be picked
either and the number of the goats needs to be reduced
further by the number of opened doors (o) to
g
o1


1d
1do
.
Furthermore, also because one cannot pick any doors
being opened to reveal a goat, the number of doors
available for being picked () needs to be reduced as
well by the number of opened doors (o), namely to
CC
i
C has already been determined to be
g
d(§4.6.4).

. The probability of
s
C will be
g
1
1
o
do

 as a
ratio of goats to doors.
The combined probability of is CC will be
g
1
1
go
dd o




. In the case of the Monty Hall prob lem, the
probability in question is as follo ws.
2211 20200
3311 31 3




This probability is zero. There are 2 goats. It is not
possible to pick a door hiding a goat by switching doors
if one has already initially picked a goat because the
other door hiding a goat is opened and the second goat is
no longer available for picking.
4.6.6. The Combined Probability of All Digital
Combinatio n E vents: 1 or 100%
In a digital perspective, two classes and their supple-
ments partition the un iverse or all that on e cou ld possibly
think about into four combination classes (§2.6). The
universe that is being considered in any partition often
exhibits unspoken or explicit limits and is therefore not
quite all that one could think about (§2.7). In a quantita-
tive perspective, the numerical probabilities of all four
combination events must add up to 1 or 100%. One or
the other of the four combination classes must be the
case and the four combination classes may take place
with different degrees of probability. Accordingly, the
following Equation ap plies. The notation is Boole’s.
1
111
11
1
cccgog c
dd odd odd o
gg o
dd o





 

(13)
Because probability exhibits simultaneously a digital
or non-quantitative facet and a quantitative facet, this
Equation can be read in two different ways (§4.6.1). If
“×” is interpreted as the Boolean AND-operator, the
reading is digital and the four terms are combination

150 L. DEPUYDT
classes of events, each characterized by the joint occur-
rence (×) of two events. If “×” is interpreted as the mul-
tiplication sign of quantitative mathematics, the four
products (×) represent the numerical probabilities of the
occurrence of each of the combination events.
4.7. The Probability of the Two Digital
Combination Events Producing the Desired
Outcome, Getting the Car, by Switching
Doors, namely and
is
CCis
CC
The Monty Hall problem and any extensions thereof re-
volve around a challenge with a desired outcome, namely
getting a car. The question is whether the desired out-
come can be achieved with greater or lesser probability
by switching doors. The focus is therefore on determin-
ing the probability of all the cases in which one gets the
car by switching doors. It appears that only two of the
four possible combination events discussed above (§4.6)
have getting a car by switching doors as the outcome,
namely is
and CCis (§§4.6.2 and 4.6.4). The
combined probab ility of these two combination ev ents is
as follows (§§4.6.2 and 4.6.4).
CC
1cc
ddo


11
g c
dd o


10c
(14)
If there is only 1 car, then and therefore
10
1do


c and 10
1
cc
ddo


 . The probability of
getting a car by switching is then as follows.
1
gc
dd o



The two products making up expression (14) have the
same denominator. Expression (14) can therefore be re-
written as follows. 1
1
cc gc
dd o




And, in light of the common factor c, also as follows.
1
1
cc g
dd o




And furthe r as fol lo w s.
1
1
cc g
dd o


cgd


And since , the following expression is also
equivalent. 1
1
cd
dd o


This is the probability that switching doors will pay off
. Extension and Generalization of the of
.1. “Doubling” the Monty Hall Problem
ery many are the ways in which the Monty Hall prob-
e than one door is
pi
.2. The Increase in Digital Complexity
ecause probability is a phenomenon exhibiting both
e case in which 2 doors are picked initially,
as
by getting a car, as stated in Equation (4) in §4.1.
5Monty Hall Problem to Any Number
Picked Doors (p)
5
V
lem can be extended. One might for example consider
cases in which one can pick three or more types of things
hiding behind doors, cars and goats and other types of
things. Exploring more of these extensions remains de-
sirable. But what can be done within the confines of the
present paper is limited. Moreover, a principal aim of the
present paper is highlighting the digital component of the
analysis of the problem and its extensions in its relation
to digitality as a fundamental component of human cog-
nition. A discussion of countless extensions of the Monty
Hall problem would probably not shed much additional
light on the fundamental assumption made here as a
working hypothesis, namely that rational thought and
language is profoundly digital. In what follows, the
analysis of extensions will be limited to cases in which
more than one door can be picked.
One specific case in which mor
cked involves “doubling” the Monty Hall problem in
all its characteristics. Accordingly, there would be 2 cars
instead of 1, 4 goats instead of 2, 6 doors instead of 3,
and 2 doors would be opened to reveal goats instead of 1.
Furthermore, the question would now be: If one picks 2
doors that remained closed and 2 doors are opened to
reveal goats, should one switch to the 2 remaining un-
opened doors to improve one’s chances of getting 2 cars
and, if so, by how much would one improve one’s
chances?
5
B
digital or non-quantitative and quantitative properties
(§§3.1 and 4.6.1), making the number of picked doors
into a variable will add complexity to the analysis of the
extended Monty Hall problem. As regards the digital
complexity, the original Monty Hall problem and the
extensions to any number of doors, cars, and opened
doors discussed above involve only digital combination
events. The additional extension to any number of picked
doors involves digital combinations of digital combina-
tion events.
Consider th
in the “doubling” of the Monty Hall problem (§5.1).
In the original Monty Hall problem and its extension to

L. DEPUYDT
151
lly picked and 2
do
ing,
tw
any number of doors, cars, and opened doors discussed
above, the initial pick of a door and the pick of a door by
switching doors are both single events.
By contrast, when 2 doors are initia
ors are picked after switching, both the initial pick an d
the pick by switching are composite. Both consist them-
selves of combination events. There are 2 initial picks
that follow one another in sequence and 2 picks by
switching that also follow one another in sequence.
In both the 2 initial picks and the 2 p icks by switch
o classes of events and their supplements generate four
digital combination classes (§2.6). As regards the 2 ini-
tial picks, the two classes of events are picking a car (C)
in the first (
f
) initial (i) pick (
f
i
C) and picking a r
(C) in the second (ca
s
) nitial ( iick (i) p
s
i
C). The two
supplement classes are failing to pick a c(ar C) in the
first (f) initial (i) pick (
f
i
C), or picking a goatnd fail-
ing to pick a car (, a
C) ine second ( th
s
) initial (i) pick
(
s
i
C), or picking a goat. Accordingly, the four digital
cobination classes characterizing just the 2 initial picks
are as follows:
m
f
isi
CC,
f
isi
CC,
f
isi
CC, and
f
isi
CC.
gardAs res the 2 picks by switching doors after doors
have been opened to reveal goats, the two classes of
events are picking a car ( C) in the first (
f
) pick by
switching (
s
) (
f
s
C) and picing a car (C) inthe second
(k
s
) pick byswiing ( tch
s
) (
s
s
C). The splement of the
first event is failing to pick ar (up a cC) in the first (
f
)
pick by switching (
s
) (
f
s
C), or pickg a goat. The su-
plement of the secot is failing to pick a car (
in p
ennd evC)
in the second (
s
) pick by switching (
s
) (
s
s
C), or pi-
ing a goat. Accordingly, the four diital mbination
classes characterizing the 2 picks by switching doors are
as follows:
ck
cog
f
sss
CC,
f
sss
CC,
f
sss
CC, and
f
sss
CC.
.3. The 16 Digital Combinations of the
ach of the four combination events of the in itial pick s is
ars p = doors picked in the initial picks e
5“Doubled” Monty Hall Problem
E
combined with each of the four combination classes of
the picks by switching. The result is 16 combinations of
combination events. The 16 combinations are listed be-
low along with the numerical probability of their occur-
rence.
c = c
g = goats o = doors opened to reveal a goat after th
intial picks
(1)

:
12
fi sifs ss
CC CC
cc c
dd dpod







3
0
11
c
po
 




(2)




:
12
0
11
fi sifs ss
CC CC
cg cc
dd dpodpo





 
 



(3)




:
12
0
11
fi sifs ss
CC CC
gc cc
dd dpodpo





 





(4)




:
11
11
4321 6
6521 15
fi sifs ss
CC CC
ggcc
dd dpodpo

 

 



 






(5)




:
12 0
11
fi sifs ss
CC CC
cc cgo
dd dpodpo

  


 
 



(6)




:
11
11
24112
6521 15
fi sifs ss
CC CC
cg cgo
dd dpodpo




 



 






(7)




:
11
11
42112
6521 15
fi sifs ss
CC CC
gc cgo
dd dpodpo




 



 






(8)




:
12
0
11
fi sifs ss
CC CC
ggc go
dd dpodpo

 


 





(9)




:
12
0
11
fi sifs ss
CC CC
ccgo c
dd dpodpo

 


 






152 L. DEPUYDT
(10)

:
1
24112
6521 15
fi sifs ss
CC CC
cg go
dd dpo

 


 







11
1
c
dpo





(11)

:
42112
6521 15
fi sifs ss
CC CC
gc go
dd dpo













11
11
c
dpo






(12)

:
12
1
fi sifs ss
CC CC
gg go
dd dpod




 




0
1
c
po





(13)

:
1
2121 1
6121 15
fi sifs ss
CC CC
cc go
dd dpo













11
1
go
dpo
 



(14)

:
1
fi sifs ss
CC CC
cg go
dd dpod





 


12
0
1
go
po

 



(15)

:
fi sifs ss
CC CC
g cgo
dd dpod




 
 


12
0
11
go
po

 



(16)

:
12
1
fi sifs ss
CC CC
gg go
dd dpod



 
 



3
0
1
go
po






probability of 10 combinations of combination
events is zero. In (1), (2), (3), (5), and (9), cars are
picked in 3 or 4 of the 4 picks, but there are only 2 cars.
In (8), (12), (14), (15), and (16), goats are picked i3 or
4 of the 4 picks, but it is not possible to pick more than 2
in total. There are two scenarios. First, in (8), (12), and
(1
rd on the first two, and th e fourth on the first
ree. For example, in (4), a door hiding a go at is in itially
ent. The chance of picking a goat is
The
n
6), 2 goats are initially picked. When 2 doors are then
opened to reveal goats, there are no goats left to pick.
Yet, 1 goat is picked according to (8), (12), and (16).
This is not possible. Second, in (14) and (15), 1 goat is
initially picked. When 2 doors are then opened to reveal
goats, only 1 goat is available for picking. Yet, 2 goats
are picked according to (14) and (15). This is also not
possible.
5.4. Dependence
Each combination in §5.3 consists of a sequence of four
events, with the second event being dependent on the
first, the thi
th
picked in the first ev
cd. In the second event, a door hiding a goat is again
picked. But the number of the goats and the doors has
been reduced by 1 in the fi rst event. The chance of again
picking a goat in the second event is therefore




11cd

. In the third event, a car is pick ed. But the
number of doors has been reduced by the 2 doors that are
ed in the first and second events (p), though not the
number of cars as no car has been picked yet. In addition,
the number of doors is reduced by 2 as 2 doors are
opened to reveal a goat (o). The chance of picking a car
pick
is therefore




2cdpo

 . Finally, in the fourth
event, a car is again picked. But, the number of cars has
been reduced by 1. So has again the number of doors.
The chance of picking a car is therefore




31cdpo

. When goats are picked by swit-
ching doors, that is, in the third and fourth component
events, the n be reduced not only by
goats that are picked but also by goats revealed by open-
ing doors (o).
y of a Successful Outcome by
Switching Doors in the “Doubled” Monty
Hall Problem
Is one more likely or less likely of getting 2 cars by
switching doo rs
umber of goats can
5.5. The Probabilit
than by not switchi n g do ors after 2 doors
goat
abili 2 cars by not switching and by
switching need to be compared.
have been opened to reveal goats if there are 2 cars, 4
s, and 6 doors? To answer this question, the prob-
ties of getting the
The probabilit y of getting the 2 cars by not switching is
the same as the probability of initially getting 2 cars by
picking 2 do ors. The pro bability o f pickin g 1 of th e cars at
first pick is cd or 26 or 13. If one was successful
in picking a car at first pick, then not only the number of
the doors but also the number of the cars is reduced by 1.
The probability of picking a car again at second pick is
therefore




11cd

 or




21 61 or 15. The
probability of picking both cars in the first 2 picks, then,
is therefore 13 15

or 115 r about 6.6%.
As for the probability of getting the 2 cars by switching
o

L. DEPUYDT
153
ars are pickednly 1 of the 1ita
ce of th
from the 2 doors initially picked to the 2 only doors that
remain closed after 2 doors have been opened to reveal 2
goats, 2 c in o6 digl com-
binations listed in §5.3, namely (4). The probability of
the occurrenis coation is mbin615 or 25 or
4
s
g it after switching doors. The
robability of gettine car is
0%. Remarkably, one can only get the 2 cars by swit-
ching doors if one had originally picked 2 doors hiding
goats or no cars at all.
5.6. Comparison of the Original and the
“Doubled” Monty Hall Problem
In the original Monty Hall problem, one is more likely to
get the car than not gettin
g thp23. By contrast, in
et the 2
cars doors. The
robability of g is
“doubling” the problem, one is less likely to g
than not getting them after switching
petting them615. On the other hand,
in the original Monty Hall problem, one doubles one’s
chances to get the car from 13 to 23 by switching
doors. By contrast, in “doubling”e problem, one’s
chances of getting both cars increase sixfold from
th115
to 615.
5.7. Compressing Probability as an Effect of
Opening Doors to Rev Goa
As always, opening doors to reveal goats has the e
of press
ealts
ffect
coming or condensing a greater robability to
wer doors (§3.3). There is a probf only
p
ability ofe115
5.5).
How
of getting both cars by initially picking 2 doors (§
ever, there is a probability of 615 or 40% th at the
cars are both hiding behind 2 of the other 4 doors that
rs 1
nd 4
as
minator
ll 16 digital combination events listed in §5.3 exhibit
mber of
pick
2
one has not picked. This number is obtained as follows.
There are 15 different ways in which the 2 cars can be
hiding behind the 6 doors: behind doors 1 and 2, doo
and 3, doors 1 and 4, doors 1 and 5, doors 1 and 6, doors
2 and 6, doors 2 and 3, doors 2 a, doors 2 and 5,
doors 2 and 6, d oors 3 and 4, doors 3 and 5, door s 3 and
6, doors 4 and 5, doors 4 and 6, and doors 5 and 6. Let us
sume that the doors picked initially are 1 and 2. In 6
out of a total of 15 ways, or 40%, the 2 cars are hiding
behind the 4 other doors, namely doors 3 to 6. They are
the last 6 locations in the list just provided. In 9 of 15
ways, at least 1 car is hiding behind either door 1 or door
2 or both.
5.8. Denominator and Numerator of the
Probabilities Involved in the “Doubled”
Monty Hall Problem
5.8.1. Den o
A
the same denominator. Furthermore, as the nu
s increases, the denominator will morph into






123d ddd
 

1
3
dp odp o
po
 
2dp o d



as fol-
lows.
 
This progression is obviously factorial, as expressed by
“!”. For any number of doors (d), opened doors (o), and
picked doors (p), the numerator will therefore be



!
!dpo d
!
dp dpo p  




!
The details exceed the scope of the present paper. Suf-
fice it to note that 0!, which is obtained when


0o pdp

, is not the same as zero or nothing
but rather signifies the absence of any additional factor.
5.8.2. Numera tor
As of what
is sought. The que
g doors. But what if one took satisfaction with getting
s one improve one’s chances in that
s regards the numerator of the 16 combination classes
listed in §5.3, an evaluation is necessary in term
stion was asked before how much one
improves one’s chances of getting both doors by switch-
in
just 1 car? How doe
case by switching both doors? One’s initial chances of
getting at least 1 car by picking 2 doors is the combined
probability of the three combined events of picking first
a car and then again a car, of picking first a car and then
a goat, and of picking first a goat and then a car, or
1
111
ccc gg c
dd dddd




, in this case
212442 9
65656 515

 . As regards switching, the
outcome of five of the 16 combination classes listed
above includes getting at least 1 car. They are combina-
tion classes (4), (6), (7), (10), r combined
probability is and (11). Thei 14 15. In sum, one does not double one’s
chances of getting at leas
the probability of not getting one has been reduced to as
t 1 car by switching doors. But
little as 115.
6. Conclusions
Walking in the footsteps of Boole has made it possible, I
believe, to construct a map of the mathematical structure
of the M Hontyall problem in its context and of certain
s thereof, ev en if the analysis
rmutations remains desirable. Two
mponents, one digital-mathematical or non-quantitative
extensions or generalization
of even higher pe
co
and the other quantitative, complement one another to
make up this map. But the focus of the present paper is
not only on this mathematical structure but also on its

L. DEPUYDT
154
ng Professor
f Finance at Tshinghua University’s School of Ec
r reading and comme
n the present paper. In explaining the Monty Hall prob-
a lecture entitled “How the Bio-
lo
to, in the
fr
] J. Gill, “Bayesian Methods,” International Encycloped
ndica, Vol. 65, No. 1, 2010, pp. 57-71.
relation to the presumed digital nature of cognition as
expressed in rational thought and language.
7. Acknowledgements
I thank Dr. Michael R. Powers, Professor of Risk Man-
agement and Insurance at Temple University’s Fox
School of Business and Distinguished Visiti
o o-
nnomics and Management, foting
o
lem to students, Professor Powers prefers to have re-
course to the method known as conditional probability.
But he also believes that different approaches may sup-
plement one another.
I am grateful to Prof. Dr. Andreas Manz, Head of Re-
search at the Korean Institute of Science and Technology
(KIST) at the University of Saarbrücken, Germany, for
inviting me to participate in a workshop held at KIST on
June 30-July 2, 2010 (see www.humandocument.org ). At
this workshop, I read
gical Brain Reasons: The Four Digital Operations Un-
derlying All Rational Language and Thought.” The lec-
ture concerned the digital analysis of rational thought
and language. The ideas presented therein have inspired
the present paper on the Monty Hall problem.
I was also grateful for the opportunity to be able to
present some ideas on the digitalization of the analysis of
rational tho ught and language on April 28, 2011 in a lec-
ture entitled “The Inevitable Digitization of Language
Analysis” and read in the Department of Near & Middle
Eastern Civilizations of the University of Toron
amework of an Information and Discussion Session on
the topic “Does It All Add Up? Quantitative Reasoning
(QR) in the Humanities.”
Finally, I thank two anonymous reviewers of an earlier
version of this paper for their critical and penetrating
comments. These comments have necessitated a com-
plete overhaul of the paper.
8. References
[1] J. Rosenhouse, “The Monty Hall Problem: The Remark-
able Story Behind Math’s Most Contentious Brainteaser,”
Oxford University Press, New York and Oxford, 2009.
[2 ia
of Statistical Science, Springer, Berlin and New York,
2010, pp. 8-10.
[3] R. D. Gill, “The Monty Hall Problem Is not a Probability
Puzzle (It’s a Challenge in Mathematical Modelling),”
Statistica Neerla
doi:10.111/j.1467-9574.2010.00474.x
[4] J. S. Rosenthal, “Monty Hall, Monty Fall, Monty Crawl,”
Math Horizons, Vol. 16, No. 1, September Issue, 2008
pp. 5-7. ,
mbols for “and” (+) and “or” (–), and for
sey, 2008, pp. 93-95.
lgebra der Logik
er)seits,’ und ‘selbst’ als digitales Wortfeld im Ägyp-
2nd edition, 1986, p. 215.
[5] I. Grattan-Guinness, “The Search for Mathematical Roots,”
Princeton University Press, Princeton and Oxford, 2000.
[6] “The sy
closed-“true” (0) and open-“false” (1) as used in switch-
ing algebra, are the inverse of the conventions used in the
algebra of logic. However, because of the perfect duality
inherent in the algebra, the postulates and theorems are
not affected” (W. Keister, A.E. Ritchie, S.H. Washburn,
“The Design of Switching Circuits,” The Bell Telephone
Laboratories Series, D. Van Nostrand Company, New
York, Toronto, and London, 1951, p. 70 note *).
[7] M. Helft, “Google Can Now Say No to ‘Raw Fish Shoes’
in 52 Languages,” The New York Times, March 9, 2010,
pp. A1 and A3.
[8] L. Depuydt, “The Other Mathematics: Language and
Logic in Egyptian and in General,” Gorgias Press, Pis-
cataway, New Jer
[9] J. Venn, “Symbolic Logic,” 2nd Edition, Macmillan and
Company, London and New York, 1894, pp. 245-255.
[10] E. Schröder, “Vorlesungen über Die A
(Exakte Logik),” J. C. Hinrichs, Leipzig, Vol. 1, 1890, p.
319.
[11] L. Depuydt, “Zur Unausweichlichen Digitalisierung der
Sprachbetrachtung: ‘Allein,’ ‘anderer,’ ‘auch,’ ‘einziger,’
‘(sein
tisch-Koptischen und im Allgemeinen” (“On the Unavoid-
able Digitalization of Language Analysis: ‘Alone,’
‘Other,’ ‘Also,’ ‘Only,’ ‘On (his) part,’ and ‘Self’ as a
Lexical Field of Digital Purport in Egyptian-Coptic and
in General”), to appear in the series Aegyptiaca Monas-
teriensia as part of the acts of the Workshop “Lexical
Fields, Semantics and Lexicography” held 5-7 November,
2010 at the University of Münster, Germany.
[12] L. Depuydt, “The Other Mathematics: Language and
Logic in Egyptian and in General,” Piscataway, New Jer-
sey, 2008, pp. 285-306.
[13] Th. Hailperin, “Boole’s Logic and Probability,” North-
Holland Publishing Company, New York and Oxford,
1976, p. 131. Also see the