Open Journal of Discrete Mathematics, 2012, 2, 164-168
http://dx.doi.org/10.4236/ojdm.2012.24033 Published Online October 2012 (http://www.SciRP.org/journal/ojdm)
Clarifying the Language of Chance Using Basic
Conditional Probability Reasoning: The Monty Hall
Problem
Pejmon Sadri
Department of Liberal Arts, Menlo College, Atherton, USA
Received July 21, 2012; revised August 2, 2012; accepted August 14, 2012
ABSTRACT
Clarity and preciseness in the use of language is crucial when communicating mathematical and probabilistic ideas.
Lack of these can make even the simplest problem difficult to understand and solve. One such problem is the Monty
Hall problem. In the past, a controversy was stirred among professional mathematicians when trying to reach a consen-
sus on a solution to the problem. The problem still creates confusion among some of those who are asked to solve it for
the first time. We purport to demonstrate the use of more precise language of basic conditional probability could have
prevented the controversy.
Keywords: Conditional Probability; The Monty Hall Problem
1. Introduction
Some loss in the meaning of expressions is expected when
information is translated from one language to another.
But, a loss in meaning isn’t the worst byproduct of
translating information. Rather, it’s the misconceptions
that are created during the process of conveying a mes-
sage from one abstract form to another that is the bigger
issue [1].
The implication of this for teaching and learning ma-
thematics is significant because a fruitful discussion of
mathematical concepts requires facility in simultaneous
comprehension of various symbolic representations; Greek
letterings, matrices, and asymptotic charts—and above
all—one’s native tongue that has to connect to all that.
Abouchedid and Nasser [2], for example, discuss how
communicating information via a web of symbolic,
graphical, numeric, and verbose representations can cre-
ate faulty conceptions in mathematics. They reason that
the “structural differences” among these various modes
of communication require translation through “connec-
tive cognitive path s” which can ultimately br ing about an
erroneous conception of the original idea.
It can get complicated. Language of any type may
have more limitations for a flawless delivery of meaning
among interlocutors than we may be aware of. The phi-
losopher Wittgenstein [3] said “If there did not exist an
agreement in what we call ‘red’, etc., language would
stop”. At the same time, he raises a more important poin t
by asking “But what about the agreement in what we call
‘agreement’?”
2. Precise and Imprecise Use of Language
Indeed, sometimes, a simple lack of atten tion to what we
have agreed on about the meaning of words and phrases
can create controversy, or a fair amount of confusion,
within the same—not different—form of linguistic rep-
resentation. One notable example of this is known as the
Monty Hall Problem, also known as “The Monty Hall
Dilemma”. The problem was originally conceived in the
1970s. A description of it is provided below [4]:
A contestant in a game show is given a choice of three
doors. Behind one is a car; behind each of the other two,
a goat. She selects Door A. However, before the door is
opened, the host opens Door C and reveals a goat. He
then asks the contestant: “Do you want to switch your
choice to Door B?” Is it to the advantage of the contest-
ant (who wants the car) to switch?
A controversy broke out among career mathematicians
and statisticians over whether or not it would be advan-
tageous for the contestant to switch his or her original
choice of door. One side in the controversy argued that
once the game show host opens one of the doors behind
which there is a goat, there will be two closed doors left
behind one of which there could be the car, and therefore,
there is only a 50/50 “chance” that the contestan t will wi n
the car, irrespective of which door he or she stays with.
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P. SADRI 165
The other side in the controversy, of course, disagreed
with this view. They argued that the contestant’s best
chance for winning the car was only 1/3 if he/she stayed
with the original choice, whereas if the contestant switched
from the original choice to the door that the host did not
open, the “odds” of winning the car would always in-
crease to 2/3. This latter answer is, of course, the correct
answer. One good and clear justification in support of
this answer can be given using basic conditional prob-
ability. The tree diagrams in Figures 1 and 2 depict this
approach. Figure 1 shows all possible outcomes for any
choice that the contestant and host might make.
Note that Grinstead and Snell [5] use numbers for la-
beling doors in the tree diagram, whereas Barbean [4]
uses English alphabet in his description of the problem.
Now suppose as an example that the contestant chooses
door 1 and Monty opens door 2. Given this scenario, only
two path outcomes from Figure 1 are possible wh ich le ad
to two conditionally probable outcomes. These are shown
in the Figure 2 diagram.
At the beginning of the contest, and before any choice
is made by the contestant, the following individual prob-
abilities apply to the car being behind any of the doors,
which we will refer to as 
1
P
D, and


3
P
D. ,


2
P
D





123
13, 113D PD
(1)
Once the contestant picks door 1 to win, the pr
lit
3, PDP
obabi-
y of door 1
D having the car behind it still stands at
1/3, but given tis choice by the contestant, the group of
two doors 2
D and 3 D now has con ditional probability
2/3 of contag the c
h
nini ar.




12
13, PD PD
3
1 31 323D (2)
Note that using the langu age of cond itional prob
w
ability
e can easily show that it is always better for the con-
Pla cemen t
of CarDoor chose n by contestantDo or opened by MontyPath probab ilities
2
3
3
2
2
2
1
3
3
1
1
1
1
2
3
1/3
1/3
1/3 1
1
1
2
2
2
3
3
3
1/3
1/3
1/3
1/3
1/3
1/3
1/3
1/3
1/3
1
1
1
1
1
1
1/2
1/2
1/2
1/2
1/2
1/2
1/18
1/18
1/9
1/9
1/9
1/18
1/18
1/9
1/9
1/9
1/18
1/18
Figure 1. Tree diagram for the Monty Hall Problem [5].
Placem ent
of carDoor c hose n by contestantDo or opene d by MontyPath probabilities Conditional probabilities
1
3
12
1 2
1/3
1/3
1/3
1/3
1/2
1
1/18 1/3
1/9 2/3
Figure 2. Tree diagram of conditional probabilities when the contestant chooses door 1 and Monty opens door 2.

P. SADRI 166
estantt to switch from the original selection because by
switching, he/she will increase th e probabilit y of winn ing
the car by entering the group that has the higher likely-
hood of con taining it. And when the ho st opens door D2,
the probability for the entire group will now belong to,
and falls under, door D3. That is:



32
PD PD
3
23D (3)
Although the answer to the Monty Hall p
co
I’l
ough
roblem be-
mes rather obvious using basic conditional probability,
this was not the approach by those arguing their case in
the controversy over the Monty Hall problem. Both sides
in the controversy used much less preciser, and some-
times casual language, to state their thinking. Some used
the word “probability”, some used the word “odds”,
some used the word “chance”, and yet others used a mix
of these simultaneously as if all these terms refer to the
same thing. But did such an agreement, as Wittgenstein
would have probably wondered, exist between the parties
to the controversy that all these terms mean the same
thing? The possibility of that being the case seems not
only remote but also highly unlikely. Hence, faulty rea-
soning goes undiscovered and communication itself be-
gins to become ineffective—as Wittgenstein had sug-
gested. Consider the following examples of what was
said:
Marilyn vos Savant [6], whose answer to the problem
agrees with Grinstead and Snell [5], says “Yes; you
should switch. The first door has a one-third chance of
winning, but the second door has a two-thirds chance.”
vos Savant’s use of the casual word “chance” bypasses
the clarity of the conditional probabilistic nature of the
solution.
vos Savant received many rebuttals with regard to her
answer to the Monty Hall Problem. The following two
quotations are examples of the responses she received,
which were included in her column published in the De-
cember 2 issue of Parade Magazine [7] that same year:
You answered, “Yes. The firs t door ha s a 1/3 chance of
winning, but the second has a 2/3 chance.” Let me ex-
plain: if one door is shown to be a loser, that information
changes the probability of eith er remaining choice to 1/2.
As a professional mathematician, I’m very concerned
with the general public’s lack of mathematical skills.
Please help by confessing your error and, in the future,
being more careful—Robert Sachs, Ph.D., George Ma-
son University, Fairfax, Va.
Again, in this rebuttal to vos Savant’s response, the
conditional probabilistic nature of the problem has been
overlooked. Let’s look at another reply to vos Savant
from the same issue of the Parade Magazine [7].
You blew it, and you blew it big! Since you seem to
have difficulty grasping the basic principle at work here,
l explain: After the host reveals a goat, you now have a
mathematical illiteracy in this country, and we don’t
need the world’s highest IQ propagating more. Shame!—
Scott Smith, Ph.D., University of Florida.
In this response, the commentator prefers to use the
word “chance” casually as vos Savant herself did, with
yet again, the language of conditional probability having
be
one-in-two chance of being correct. There is en
en overlooked in the explanation.
Moreover, what is even more interesting is that in her
own rebuttal to the rebuttals, vos Savant [7] uses the
language of “odds” rather than the language of “chance”
which she had initially used:
Let me explain why your answer is wrong. The win-
ning odds of 1/3 on the first ch o ice ca n’t go up to 1/2 just
because the host opens a losing door. To illustrate this,
let’s say we play a shell game. You look away, and I put
a pea under one of the three shells. Then I ask you to put
your finger on a shell. The odds that your choice con-
tains a pea are 1/3, agreed? Then I simply lift up an
empty shell from the remaining two. As I can (and will)
do this regardless of what you’ve chosen, we’ve learned
nothing to allow us to revise the odds on the shell under
your finger.
This is remarkable! Even though, vos Savant has the
right ideas and the answer to the problem, she is not
aware of her own inconsistent use of the terms when ex-
plaining her own thinking. She uses the word “odds” in
the December issue of the magazine, whereas in her re-
sponse given in the September issue, she uses the word
“chance”. Clearly, there does not seem to be an explicit
or underlying agreement between the debaters about a
single probabilistic concept that could be used as repre-
sentation of probabilities before and after the host opens
a door. No wonder there was confusion and controversy
over the correct answer to a simple problem of probabil-
istic nature. In fact, a cursory look through introductory
texts in probability tells us that although “odds” and
“probability” are closely related, they are not one and the
same concept. Conceptually, they are described as:
N
o. of desired outcomes
Probability,
N
o. of possible outcomes
No. of desired outcomes

(4)
OddsNo. of undesired outcomes

And each can be written in terms of the other a
lows [8]: s fol-
Odds
Probability,
1 Odds (5)
Probability
Odds 1 Probability


More formally, if a statement postulates that the odds
are to r
s
in favor of an outcome E occurring, then
the probability of the outcome in terms of the given odds

P. SADRI 167
can be given as [5]:

r
PE rs 
From the different answers provided to the Monty Hall
pr
ding the solution.
Thankfully, the casual and incon
abilistic terms is not unanimous among professionals as
th
to poker players
bu
s” within a spe-
ci
oint. And it was once again another quantum
ph
to the
us
n dis-
riting. For example,
(6)
oblem by the mathematicians, it is clear that they con-
ceptualizedthe problem differently. At the same time, a
lack of consistent, precise, accurate, and clear use of lan-
guage of probability seems to have complicated reach-
ing a consensus regarsistent use of prob-
ere are those who make every effort to avoid vagueness.
For example, Matthew Hilger [9], an expert of rules and
strategies used in gambling, makes a point of being clear
by saying that “Probabilities tell you how frequently an
event will happen, odds tell you how many times an
event will not happen”. His advice
ilds on that distinction. “To determine the odds against
improving your hand on the next card,” he writes, “com-
pare the total number of cards that will not help you to
the number of cards or ‘outs’ that will.”
To use another example, Charles Darwin [10] was
careful when discussing uncertainty in his seminal work,
On the Origin of Species. He used the word “probabi-
lity” almost exclusively when discussing adaptation of
species over very long periods of time and many hun-
dreds of generations. On the other hand, he used the
word “chance” almost exclusively when speaking about
survival of species, “profitable variation
es, and also when referring to propagation of attributes
from parents to their off-spring. Darwin never used the
word “odds” in the context of survival or adaptation of
species. This demonstrates how careful and conscious he
was about applying probabilistic terms to distinguish
between short-term and long-term trends in biological
evolution.
Similarly, we see the same level of care and deliberate
attention in the use of probabilistic language among phy-
sicists who dedicate themselves to the study of subatomic
particles. The Nobel Prize winner Max Born [11] ob-
served that it was “the concept of probability that was
applied systematically and built into the system of phys-
ics”. Note that he didn’t use the word odds or chance to
make his p
ysicist and renowned String Theorist, Brian Greene
[12], who wrote “We are accustomed to probability
showing up in horse races, in coin tosses, and at the rou-
lette table,” he said, “But in those cases it merely reflects
our incomplete knowledge.” Once again, note how con-
scious Greene is abou t the specific meaning that the term
probability might take on within various contexts.
The Monty Hall problem presents a far less compli-
cated situation than trying to discern between biological
features that are the result of genetic transmission rather
than adaptation of a species over thousands of years, or
the situation of trying to predict the most likely location
of an electron on the electron wave in quantum mechan-
ics. Do we know for sure that it is only carelessness on
the part of professional interlocutors with respect
e of the probabilistic terms odds, probability, or chance
that fueled the controversy over the Monty Hall problem?
No. However, it is clear from looking at published argu-
ments from both sides that neither side arguing the case
seemed to be aware of the sensitivity of the issue of using
the right probabilistic language to make their case.
3. Conclusions
As Wittgenstein poin ted out, it is the fundamental agree-
ment about the de finition of words, which we internalize
growing up as a child and into adulthood, that makes
language work the way it does. Teachers of mathematics
and statistics often use notation to establish a consensus
about similarity or difference among concepts whe
and
x
cussing them in w

are
etween population and sample means,
p
nvironment. More important than what mathe-
m
[3] L. Wittgenstein,
matics,” Revisambridge, 1983.
154.
used to distinguish b
respectively. Because dissemination of information or
exchanging of ideas through sp oken and written words is
a major part of any form of education, care must be taken
to avoid posing dissimilar concepts as similar or vice
versa.
The ability to have clarity and deliberate precision in
the use of language is a valuable asset and can helalle-
viate misunderstanding when communicating with others;
an asset that great ones like Darwin, Born, and Greene,
for example, apparently possessed and made use of in
their literary work. But a more significant corollary of
this conclusion is that it is even more valid in the class-
room e
atical content we teach is the effort we must put to
making concepts clear for our students. This means that
as professional teachers, we must keep ourselves humble
and flexible, never assuming an infallibility in our own
teaching and communication skills.
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P. SADRI 168
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