Paradoxes and Fallacies - Resolving some well-known puzzles with Bayesian networks

Paradoxes and Fallacies

Resolving some well-known puzzles with Bayesian networks

Stefan Conrady, stefan.conrady@conradyscience.com

Dr. Lionel Jouffe, jouffe@bayesia.com

May 2, 2011

Conrady Applied Science, LLC - Bayesia’s North American Partner for Sales and Consulting


Table of Contents

Introduction
Background & Objective 1
Notation 2

Prosecutors Fallacy 3

Simpson’s Paradox 7

The Monty Hall Problem 12

Conclusion 15

Appendix
Bayes’ Theorem 16

About the Authors 16
Stefan Conrady 16
Lionel Jouffe 16

References 17

Contact Information 18
Conrady Applied Science, LLC 18
Bayesia SAS 18

Copyright 18

www.conradyscience.com | www.bayesia.com
ii


Introduction

Background & Objective
There are a number of paradoxes and fallacies that keep recurring as popular and mind-bending puzzles in the media.
Although there is (now) complete agreement among scientists on how to resolve them, the correct answers are often
perplexing to the casual observer and still cause bewilderment.

We will start off with the fallacy of the transposed conditional, which has become rather infamous and is better known
as Prosecutor’s Fallacy. As the name implies, it is a problem often encountered in courts of law and there are numerous
cases of incorrect convictions as a result of this fallacy.

No less serious are the potential consequences of Simpson’s Paradox, for instance, when determining the treatment ef-
fect of a new drug under study. The effect of a drug on two subgroups may appear as the complete opposite of the
treatment effect on the whole group.

On a much lighter note, the Monty Hall Problem has its origin in a television game show and might perhaps be the
most dif cult puzzle to comprehend intuitively, even when explicit proof is provided. Respected mathematicians and
statisticians have struggled with this problem and some of them have boldly proclaimed wrong solutions.

The counterintuitive nature of these probabilistic problems relates to the cognitive limits of human inference. More spe-
ci cally, we are dealing with the problem of updating beliefs given new evidence, i.e. carrying out inference. This cogni-
tive challenge may seem surprising, given that humans are exceptionally gifted in discovering causal structures in their
everyday environment. Discovering causality in the world is quite literally child’s play, as babies start understanding the
world through a combination of observation and experimentation. Our human intuition is actually quite good when it
comes to reasoning from cause to effect and our qualitative perception of such relationships (even under uncertainty) is
often compatible with formal computations.

However, when it comes to reasoning under uncertainty in the opposite direction, from effect to cause, i.e. diagnosis, or
when combining multiple pieces of evidence, conventional wisdom frequently fails catastrophically. Even worse, the
correct inference in such situations is often completely counterintuitive to people and feels utterly wrong to them. It is
not an exaggeration to say that their sense of reason is violated.

For more traditional computations, such as arithmetics, we have many tools that help us address our mental shortcom-
ings. For instance, we can use paper and pencil to add 9,263,891 and 1,421,602, as most of us can’t do this in our
heads. Alternatively, we can use a spreadsheet for this computation. In any case, it will not surprise us that the sum of
those two numbers is a little over 10.5 million. The computed result is entirely consistent with our intuition.

As this paper will show, the formally correct solutions of these probabilistic paradoxes are counterintuitive. In addition
to being counterintuitive, there are few tools assisting us in solving them. There is no spreadsheet that allows us to sim-
ply plug in the numbers to calculate the result.

Although we won’t be able to overcome inherent mental biases and cognitive limitations, we can now provide a very
practical new tool for the correct inference in the form of Bayesian networks. Bayesian networks derive their name from
Reverend Thomas Bayes, who, in the middle of the 18th century, rst stated the rule for computing inverse probabili-
ties.

1


Bayesian networks offer a framework that allows applying Bayes’ Rule for updating beliefs in the same way spread-
sheets are very convenient for applying arithmetic operations to many numbers. We will show how restating these vex-
ing problem domains as simple Bayesian networks offers near-instant solutions. Just as spreadsheets help us perform
arithmetic operations externally, i.e. outside our head, Bayesian networks offer a reliable structure to precisely perform
inferential computations, which we can’t manage in our minds. The visual nature of Bayesian networks furthermore
helps (at least a little) in making these paradoxes more intuitive to our own human way of thinking.

Beyond utilizing Bayesian networks as the framework, we will use BayesiaLab 5.01 as the software tool for network
creation, editing and inference. This allows us to leverage all the theoretical bene ts of Bayesian networks for practical
use via an intuitive graphical user interface.

Notation
To clearly distinguish between natural language, software-speci c functions and example-speci c variable names, the
following notation is used:

• BayesiaLab-speci c functions, keywords, commands, etc., are capitalized and shown in bold type.

• Names of attributes, variables, nodes and are italicized.

1 An evaluation version of BayesiaLab can be downloaded from
http://www.bayesia.com/en/products/bayesialab/download.php. All examples discussed in this paper can be replicated
with this trial version.

2



Prosecutors Fallacy
Crime dramas and live courtroom reporting have familiarized all of us with this situation, whether hypothetical or real:
the prosecutor calls an expert to the witness stand and queries him about the reliability of evidence found at a crime
scene. The expert, typically a physician or medical examiner, will state something like, “the probability of nding — by
chance — the blood type at the crime scene which matches the one of the defendant is about one in 1,000.” The prose-
cutor will presumably be satis ed with this answer and probably paraphrase it in his closing argument to jury: “as you
can see, the there is only a one in 1,000 chance that the defendant is innocent and therefore it is clear beyond any rea-
sonable doubt that the defendant is guilty.”

It wouldn’t be the Prosecutors Fallacy if there wasn’t a problem with this seemingly plausible conclusion. So, what’s
wrong? Let us restate the expert witness’ testimony and furthermore clarify some implicit assumptions:

“The probability of identifying (or matching) some innocent person’s blood type at a crime scene by chance (or sheer
coincidence) is one in 1,000.” This is equivalent to the following:

1
P(Match=true Crime=false) = = 10 −3
1, 000

In words, given that someone has not committed the crime, there is a 1/1,000 chance of identifying his or her blood type
at the scene of the crime by sheer coincidence.

However, the prosecutor claimed something else: “Given the evidence, there is only a 1/1,000 chance that the defendant
is not guilty,” which is a different statement:

1
P(Crime=false Match=true) = = 10 −3
1, 000

So, should the jury nd the defendant guilty? Maybe. Further assumptions are required to compute the correct probabil-
ity of the defendant having committed the crime.

The rst assumption is about the probability of a blood type match, given that one has actually committed the crime.
Let us assume that this probability is 1, i.e.

P(Match=true Crime=true) = 1

Furthermore, we need to understand the base rate of the crime. For instance, statistics might tell us that this crime hap-
pens only very rarely, e.g. only once in a city of 10,000 in a given time period. So, this is the marginal probability of
being guilty:

1
P(Crime=true) = = 10 −4
10, 000

Without any other knowledge, the probability of anyone in this city being guilty of such a crime is one in 10,000.

3


We can now use Bayes’ Rule2 to compute the probability in question, i.e. the probability of the defendant being guilty.
For a more compact representation, we will write:

“Match=true” = “Evidence”= “E”

“Crime=true” = “Guilty” = “G”

“Match=false” = “Not Evidence” = “¬E”

“Crime=false” = “Not Guilty” = “¬Guilty” = “¬G”.

The Bayes’ Rule will thus say:

P(E | G)P(G)
P(G | E) =
P(E)

The only unknown in this formula is P(E), i.e. the marginal probability of nding evidence by chance. To be more pre-
cise, we can employ the law of total probability, which in our case translates into:

P(E) = P(E, ¬G) + P(E,G) =
P(E ¬G)P(¬G)+P(E G)P(G)

We already know that

P(¬G) = 1 − P(G) ,

and hence we can compute:

P(E | G)P(G) P(E | G)P(G) 1⋅10 −4 1
P(G | E) = = = −3 −4 −4
≈ = 0.091 = 9%
P(E) P(E | ¬G)P(¬G) + P(E | G)P(G) 10 ⋅ (1 − 10 ) + 1⋅10 10 + 1

So, given the evidence of a blood type match, the defendant has a 9% probability of being guilty, which is presumably
not enough for a conviction. However, as the marginal probability of being guilty is only 0.01%, the probability of the
defendant’s guilt has risen 900-fold, given that the blood type matches.

However, the above approach may still prove to be cumbersome for practical use, especially as the real-world condi-
tions are typically much more complex. As an alternative, we can represent this problem domain as a Bayesian network
and create a network graph in BayesiaLab. In BayesiaLab, variables are represented as blue nodes and direct probabilis-
tic relationships are shown as arcs. The direction of such arcs may represent a causal assumption.

In our case, the network of the problem domain will look like this:

2 More details about Bayes’ Rule are provided in the appendix.

4


However, for now this only says, whether or not a crime has occurred will have a direct in uence on the probability of
whether or not evidence is found.

To use this Bayesian network and BayesiaLab for inference, we also need to specify all known probabilities, e.g. from
crime statistics, from the expert witness, etc. We can enter these values via BayesiaLab’s Node Editor.

This will associate a marginal distribution with Crime and a conditional probability distribution for Evidence, as illus-
trated below.

In this format, BayesiaLab can carry out inference automatically. However, prior to observing any crime or evidence,
the prior probabilities would be shown by default in BayesiaLab’s Monitor Panel.

5


In BayesiaLab, Monitors are small bar charts which display the distributions of any selected variable in the network.
For reference, the graphical user interface is shown in the screenshot below. The network and the Monitors appear in
the Graph Panel (left) and in the Monitor Panel (right) respectively.

Within BayesiaLab we can now simply carry out inference by observing evidence, i.e. by setting Evidence=“True”,

and BayesiaLab will automatically update the conditional probability distribution of Crime:

As we computed the probability of the cause given its effect, this represents a form of diagnosis.3 We have now arrived
at the same conclusion, except that BayesiaLab has performed all the necessary computations for us.4

3 The term diagnosis is more common in the medical context, where a physician may determine the probability of a
speci c illness, given certain symptoms. The direction of inference, from effect to cause, is the same though.
4 While the correctness of such probabilistic computations in BayesiaLab (and in other programs) are undisputed in the
scienti c community, they are, like these puzzles, still viewed with skepticism by the general public. It is unfortunate
that it will presumably take many more years before these computations will nd widespread acceptance and use in
court and in other areas of decision making.

6


Simpson’s Paradox
At the peak of the recent recession, Simpson’s Paradox made headlines again, as the media inundated us with countless
statistics about the condition of the economy. However, some of the statistics seemed utterly incongruent and thus un-
doubtedly generated con icting interpretations, perhaps furthering policymakers’ already diverging views.

It becomes an even more immediate problem when Simpson’s Paradox rears its ugly head in the context of medical stud-
ies, where it can suggest a false interpretation of a treatment effect.

We use an admittedly contrived example to illustrate this problem. A hypothetical type of cancer equally effects men
and women. A long-term study nds that a speci c type of cancer therapy increases the remission rate from 40 to 50%
among all treated patients (see table). Based on the study, this particular treatment is thus recommended for broader
application.

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However, when examining patient records by gender, the remission rate for male patients — upon treatment — de-
creases from 70% to 60% and for female patients the remission rate declines from 30% to 20% (see table). So, is this
new therapy effective overall or not?

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*"+ !"# $"#
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7


The answer lies in the fact that — in this example — there was an unequal application of the treatment to men and
women. More speci cally, 75% of the male patients and only 25% of female patients received the treatment. Although
the reason for this imbalance is irrelevant for inference, one could imagine that side effects of this treatment are much
more severe for females, who thus seek alternatives therapies. As a result, there is a greater share of men among the
treated patients. Given that men also have a better recovery prospect with this type of cancer, the remission rate for the
total patient population increases.

So, what is the true overall effect of this treatment? With a Bayesian network, the paradox can be easily resolved and
the effect can be computed automatically. However, to create a Bayesian network for this purpose, we rst need to
make speci c assumptions regarding causality.5

With our knowledge of the domain, we can make such causal assumptions and thus de ne a causal network. As stated
earlier, we assume that Gender has a causal effect on Remission (rather than Remission on Gender), so we de ne the
arc Gender ➝ Remission. We also assume that Treatment has a causal effect (whether positive or negative) on Remis-
sion, which translates into Treatment ➝ Remission. Finally, we have learned that Gender in uences (causes) whether or
not one would undergo Treatment, so we have Gender ➝ Treatment.

Once we have this structure, we still need to enter all the marginal and conditional probabilities we have observed. We
can do so be specifying the values via BayesiaLab’s Node Editor. The following illustration shows the network plus the
tables associated with each node. For Gender, we have a one-dimensional table (marginal probabilities only), for
Treatment, a two-dimensional table (conditional probabilities, given Gender) and nally, for Remission, a three-
dimensional table (conditional probabilities, given Gender and Treatment).

5 The concept of causality has been highly controversial over the last 100 years and for a long time it seemed entirely
banned from statistical literature. Causality has emerged from obscurity in recent decades in now plays a central role in
the study of Bayesian networks.

8


Now the structure and the parameters of the Bayesian network are de ned and we can proceed to inference. The origi-
nal statement about this domain was that, given Treatment and without specifying Gender, total Remission increases
from 40% to 50%. If this Bayesian network is a correct representation of our domain, it will need to return the propor-
tions as we observed them originally.

By setting evidence on the Treatment node and not setting evidence to Gender we can test this.

In the bottom Monitors we can now see that Remission indeed goes from 40% to 50%, but we also see that, given
Treatment, the proportion of men grows from 25% to 75% (top Monitors). This re ects the omnidirectional inference
property of Bayesian networks. Even though we were only looking for inference on Remission, we inevitably saw an-
other implication, namely, given Treatment, the balance of Gender also changes.

9


When we now set evidence to Gender, e.g. Gender=“male”, we will con rm the seemingly paradoxical result, i.e. that
Remission decreases.

For sake of completeness, we repeat this for Gender=“female”:

We have now veri ed that all the original statements are fully compatible with the network representation, but the nal
answer remains elusive, i.e. how much does treatment affect remission in general? To answer this we must make a dis-
tinction between observational inference and causal inference. This is because of the semantic difference of “given that
we observe” versus “given that we do”. The former is strictly an observation, i.e. we focus on the patients who received
treatment, whereas the latter is an active intervention. The answer to our question of the treatment effect then is infer-
ring as to what would hypothetically happen, “given that we do”, i.e. given that we force the treatment without permit-
ting patients to self-select their treatment. In the semantics of Bayesian networks, this means that there must not be a
direct relationship between Gender and Treatment. In other words, Treatment must not directly depend on Gender. In
our Bayesian network this can be done easily by mutilating the graph, i.e. deleting the arc connecting Gender and
Treatment or by xing the distribution of Gender. BayesiaLab offers a very simple function to achieve this, which is
aptly named Intervention.

By intervening on the Treatment variable (and setting Treatment=“yes”), the causal Bayesian network is modi ed as
follows:

10


Now we can observe what happens to Remission when we “do” Treatment, instead of just “observing” Treatment.

In fact, Remission decreases from 50% to 40%, given that we “do” Treatment, and so we must conclude that the new
treatment is detrimental to the patients’ health.

Outside our made-up example, e.g. in real clinical trials, such “do” conditions are achieved with controlled, randomized
experiments, which allow investigators to determine the true effectiveness of new drugs. However, the real world is not
a controlled experiment and self-selection is often inherent in many observational studies. Without being conscious of
Simpson’s Paradox, results can be easily perceived as the opposite of the truth.

11


The Monty Hall Problem
The Monty Hall Problem is a probability puzzle based on the American television game show Let’s Make a Deal, origi-
nally hosted by Monty Hall.

In her book, The Power of Logical Thinking, Marylin vos Savant
quotes cognitive psychologist Massimo Piattelli-Palmarini as saying,
“... no other statistical puzzle comes so close to fooling all the people
all the time” and “that even Nobel physicists systematically give the
wrong answer, and that they insist on it, and they are ready to berate
in print those who propose the right answer.”

Whereas some of the earlier descriptions of the game led to different
interpretations of the problem, Krauss and Wang (2003) state a fully
unambiguous and mathematically explicit version of this problem:

Suppose you’re on a game show and you’re given the choice of three doors [and will win what is behind the
chosen door]. Behind one door is a car; behind the others, goats [i.e. silly prizes]. The car and the goats were
placed randomly behind the doors before the show. The rules of the game show are as follows: After you have
chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what
is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat
behind it. If both remaining doors have goats behind them, he chooses one [uniformly] at random. After
Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your rst
choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3,
which has a goat. He then asks you “Do you want to switch to Door Number 2?” Is it to your advantage to
change your choice?

The vast majority of people intuitively believe that, when keeping their original choice, they will have a 50/50 chance of
winning, which turns out to be incorrect. There are number of explanations, which resolve this paradox and a several of
them are detailed on Wikipedia as well as on Marylin vos Savant’s website. Rather than reiterating those explanations,
we will demonstrate how a Bayesian network can quickly and simply produce the correct answer.

The game description from above can be fully expressed with the structure and the parameters of a Bayesian network.
In terms of structure, we know that only one variable is dependent on other variables and that is Monty’s choice of
which door to open. His choice is a function of (or caused by) the contestant’s rst choice and the actual location of the
valuable prize, which Monty knows. Other than that, there are no direct dependencies. As such, we can introduce two
causal arcs pointing to Monty’s Choice.6

6 In this case there can be no doubt about the direction of the arcs, as both Contestant’s Choice and Winning Door are
determined before Monty’s Choice is set. A causal arc going back in time is obviously not possible.

12


For the parameters of this network, we simply need to restate the rules of the game as marginal and conditional prob-
abilities and enter them via BayesiaLab’s Node Editor. As the prize is randomly placed, each of the three doors is
equally likely to be the prize-winning door, thus assigning a one-third probability to each door. Without knowing the
preferences of the contestant, we assume that there will also be a one-third chance of him or her picking a speci c door
(the a-priori probability of the contestant’s choice actually does not matter).

The game rules determining Monty’s Choice can be expressed in table form and is entered that way in the Node Editor:

The fully speci ed network is shown below with its associated marginal and conditional probability tables.

Let’s go through this example, and for the sake of argument assume that the contestant picks Door 1. If the prize is ac-
tually behind Door 1, Monty will pick either one of the other doors at random, so Door 2 and Door 3 will both have a
50% probability of being opened. If the prize, however, is behind Door 2, Monty will certainly not open Door 2, but
rather open Door 3, implying a 100% probability for the latter. Finally, if the prize is behind Door 3, Monty must open
Door 2. Examples for other initial door choices of the contestant follow analogously. The entire logic is fully expressed
in the conditional probability table for the node Monty’s Choice.

13


With all the parameters speci ed, we now have a Bayesian network representing our problem domain and this provides
us with our desired inference tool. We will now observe the inference progression from the contestant’s perspective as
the game evolves.

Before the game starts, all probabilities are uniformly distributed. At this point, our inference tool is of no help and the
contestant’s random pick of a door is as good as any other choice.

So, let’s assume the contestant picks Door 1, which in Bayesian network terms means setting the node Contestant’s
Choice to state 1. From the rules (and the conditional probability table) we know that Monty never opens the door the
contestant picked, so the probability of opening Door 1 is zero. As the contestant has no knowledge about the true loca-
tion of the prize, he only knows that one of the other doors will be opened and that, at this particular point in time, his
belief should be 50/50 regarding either door. This is precisely what we can see in the monitor panels, once we set Con-
testant’s Choice to state 1

Now, given his knowledge about the location of the prize, Monty responds and picks Door 2, which will inevitably
reveal a worthless prize.

14


The crucial question is, how should this observation change our contestant’s belief in the probabilities of the prize being
behind either Door 1 or Door 3? Should the contestant update his beliefs about the location of the door, given his rst
choice plus Monty’s subsequent choice?

BayesiaLab can compute this new probability distribution by setting evidence to the node Monty’s Choice, i.e. Monty’s
Choice=2, which we have just observed.

Given Monty’s choice of Door 2, BayesiaLab computes a two-thirds probability of the prize being behind Door 3 and
only a one-third probability of being behind Door 1. So, the contestant’s rational choice would be to change his choice
of doors, and not stick to his original pick.

How does BayesiaLab determine this? BayesiaLab consequently applies Bayes’ Rule to compute the new posterior prob-
abilities, given the emerging evidence. Without going into further detail, the key point is that setting evidence to Monty’s
Choice renders Contestant’s Choice and Winning Door dependent and allows information to “ ow” across nodes and
update Winning Door.

Readers, who are in still doubt, may want to experiment and repeatedly play this game, perhaps with three cups and a
coin. After several rounds, one will nd that that the probability of winning converges to a value of 2/3, if the recom-
mended switching policy is applied consistently. If not, the chance of winning remains at 1/3.

Conclusion
For reasons we have not discussed here, the cognitive skills of humans are inherently limited when it comes to dealing
with numerous pieces of evidence, especially when those piece of evidence represent uncertain observations. Bayesian
networks are a very practical tool for carrying out inference in these situations. With programs like BayesiaLab, re-
phrasing the paradoxical problem domain into a causal model is a relatively easy task, consisting of individually
intuitive steps. Given such a model, carrying out inference, which is so tricky and counterintuitive for humans, becomes
entirely automatic.

15


Appendix

Bayes’ Theorem
Bayes’ theorem relates the conditional and marginal probabilities of discrete events A and B, provided that the probabil-
ity of B does not equal zero:

P(B A)P(A)
P(A B) =
P(B)

In Bayes’ theorem, each probability has a conventional name:

• P(A) is the prior probability (or “unconditional” or “marginal” probability) of A. It is “prior” in the sense that it
does not take into account any information about B. The unconditional probability P(A) was called “a priori” by
Ronald A. Fisher.

• P(A|B) is the conditional probability of A, given B. It is also called the posterior probability because it is derived from
or depends upon the speci ed value of B.

• P(B|A) is the conditional probability of B given A. It is also called the likelihood.

• P(B) is the prior or marginal probability of B.

Bayes theorem in this form gives a mathematical representation of how the conditional probability of event A given B is
related to the converse conditional probability of B given A.

About the Authors

Stefan Conrady
Stefan Conrady is the cofounder and managing partner of Conrady Applied Science, LLC, a privately held consulting
rm specializing in knowledge discovery and probabilistic reasoning with Bayesian networks. In 2010, Conrady Applied
Science was appointed the authorized sales and consulting partner of Bayesia SAS for North America.

Stefan Conrady studied Electrical Engineering and has extensive management experience in the elds of product plan-
ning, marketing and analytics, working at Daimler and BMW Group in Europe, North America and Asia. Prior to es-
tablishing his own rm, he was heading the Analytics & Forecasting group at Nissan North America.

Lionel Jouffe
Dr. Lionel Jouffe is cofounder and CEO of France-based Bayesia SAS. Lionel Jouffe holds a Ph.D. in Computer Science
and has been working in the eld of Arti cial Intelligence since the early 1990s. He and his team have been developing
BayesiaLab since 1999 and it has emerged as the leading software package for knowledge discovery, data mining and
knowledge modeling using Bayesian networks. BayesiaLab enjoys broad acceptance in academic communities as well as
in business and industry. The relevance of Bayesian networks, especially in the context of consumer research, is high-
lighted by Bayesia’s strategic partnership with Procter & Gamble, who has deployed BayesiaLab globally since 2007.

16


References
Fountain, J., and P. Gunby. “Ambiguity, the Certainty Illusion, and Gigerenzer’s Natural Frequency Approach to Rea-
soning with Inverse Probabilities” (2010).

Glymour, Clark. The Mind’s Arrows: Bayes Nets and Graphical Causal Models in Psychology. The MIT Press, 2001.
Kahneman, Daniel, Paul Slovic, and Amos Tversky. Judgment under Uncertainty: Heuristics and Biases. 1st ed. Cam-
bridge University Press, 1982.
Krauss, S., and X. T. Wang. “The Psychology of the Monty Hall Problem: Discovering Psychological Mechanisms for
Solving a Tenacious Brain Teaser* 1,* 2.” Journal of Experimental Psychology: General 132, no. 1 (2003): 3–22.
“Monty Hall Problem.” http://en.wikipedia.org/wiki/Monty_Hall_problem.
Nobles, R., and D. Schiff. “Misleading statistics within criminal trials.” Signi cance 2, no. 1 (2005): 17–19.
Pearl, Judea. Causality: Models, Reasoning and Inference. 2nd ed. Cambridge University Press, 2009.
vos Savant, Marilyn. “Game Show Problem.” http://www.marilynvossavant.com/articles/gameshow.html.
———. The Power of Logical Thinking: Easy Lessons in the Art of Reasoning...and Hard Facts About Its Absence in
Our Lives. St. Martin’s Grif n, 1997.
Tuna, Cari. “When Combined Data Reveal the Flaw of Averages.” wsj.com, December 2, 2009, sec. The Numbers Guy.
http://online.wsj.com/article/SB125970744553071829.html#articleTabs%3Darticle.

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