# The Turtleback Diagram for
Conditional Probability

Donghui Yan and Gary E. Davis
Department of Mathematics and Program in Data Science
University of Massachusetts Dartmouth, MA

###### Abstract

We elaborate on an alternative representation of conditional probability to
the usual tree diagram. We term the representation “turtleback diagram” for
its resemblance to the pattern on turtle shells. Adopting the set theoretic
view of events and the sample space, the turtleback diagram uses elements from
Venn diagrams—set intersection, complement and partition—for conditioning,
with the additional notion that the area of a set indicates probability
whereas the ratio of areas for conditional probability. Once parts of the
diagram are drawn and properly labeled, the calculation of conditional
probability involves only simple arithmetic on the area of relevant sets. We
discuss turtleback diagrams in relation to other visual representations of
conditional probability, and detail several scenarios in which turtleback
diagrams prove useful. By the equivalence of recursive space partition and the
tree, the turtleback diagram is seen to be equally expressive as the tree
diagram for representing abstract concepts. We also provide empirical data on
the use of turtleback diagrams with undergraduate students in elementary
statistics or probability courses.

Keywords: Visualization; graph representation; recursive space partition; Venn
diagram.

##  1 Introduction

Conditional probability [1, 2, 3, 4] is an important concept in probability
and statistics. It has been widely acknowledged that the concept of
conditional probability, and particularly its application in practical
contexts, is difficult for students [5, 6, 7, 8, 9, 10, 11, 12] and especially
those without much background or previous training in mathematics at the
college level.

Let $A$ and $B$ be two events, then the conditional probability of $A$ given
$B$ is defined as

$\mathbb{P}(A|B)=\frac{\mathbb{P}(A\cap B)}{\mathbb{P}(B)}.$ (1)

Our experience with undergraduate students is that a major difficulty in
understanding and working effectively with conditional probability lies in the
level of abstraction involved in the concepts of “event” and “conditioning”;
see also [7].

The focus of this article is on productive visual representations for the
understanding and application of conditional probability. The significant role
of visual representation in mathematics is well-established; see, for example,
[13, 14]. While visualization is an important topic in statistics (see, e.g.,
[15, 16]), the role of visualization in statistics education or practice is
not as well documented. In particular, there is actually not much research
into productive visualization of conditional probability [17, 18]; popular
books such as [19] do not dedicate much effort to visual explanations of the
Bayes theorem. There has been some research on school student difficulties
with conditional probability [6, 8, 10, 11, 12] but much less so for
undergraduates. Our aim in discussing turtleback diagrams is to provide a
visual tool for the representation of conditional probability that may,
additionally, be used in further research on student understanding of
conditional probability.

##  2 Student difficulties in understanding conditional probability

Tomlinson and Quinn [9], in discussing their graphic model for representing
conditional probability (see Section 3.2.1), state:

> “Conditional probability is a difficult topic for students to master. Often
> counter-intuitive, its central laws are composed of abstract terms and
> complex equations that do not immediately mesh with subjective intuitions of
> experience. If students are to acquire the mathematical skills necessary for
> rational judgement, teaching must focus on challenging the personal biases
> and cognitive heuristics identified by psychologists, and demonstrate in the
> most accessible way—the power of probabilistic reasoning.” (p.7)

Documented student difficulties with conditional probability can be summarized
as one of three main types [7]:

  1. 1.

Interpreting conditionality as causality.

  2. 2.

Identifying and describing the conditioning event.

  3. 3.

Confusing $\mathbb{P}(A|B)$ and $\mathbb{P}(B|A)$.

Tarr and Jones [8] developed a valid and reliable framework for addressing
student difficulties with conditional probability, in the context of sampling
without replacement. This framework is particularly valuable in carrying out
research as to which visual representation of conditional probability is most
useful in assisting students and teachers.

##  3 Visual representations of conditional probability

###  3.1 Tree diagrams

Tree diagrams have been used by many to help understand conditional
probability. The idea of a tree diagram is to use nodes for events, the
splitting of a node for sub-events, and the edges in the tree for
conditioning. For example, Figure 1 is an illustration of conditional
probability. Node $*$ indicates the sample space $\Omega$, and we will use
them interchangeably throughout. Two possible events, either $B$ or
$\overline{B}$, may happen. This is represented by two tree nodes $B$ and
$\overline{B}$. The splitting of node $B$ into two nodes $A$ and
$\overline{A}$ indicates that, given $B$, two possible events, $A$ and
$\overline{A}$, may occur. The edges, $B\rightarrow A$ and
$B\rightarrow\overline{A}$ indicate conditional probabilities,
$\mathbb{P}(A|B)$ and $\mathbb{P}(\overline{A}|B)$, respectively.

Figure 1: The tree diagram approach for conditional probability.

Tree diagrams help many students to understand the concept of conditional
probability and apply it for problem solving, but is not so effective to many
others especially those less prepared ones. Basically, they find the following
two aspects non-intuitive. One is to represent events by tree nodes, which
usually appear as dots or small circles, but events are sets and are more
naturally represented by Venn diagram [20] type of notations. Another is the
idea to represent conditional probability by tree edges; it is hard to see any
straightforward connections of this to (1).

To address issues with the tree diagram, let us re-examine the idea of
graphical visualization. There are two important ingredients (or steps) in
visualizing an abstract mathematical concept. One is a concrete graphical
representation of the target mathematical objects. This step would offload
part of the burden of the brain by concrete graph objects, without which one
has to keep relevant abstract mathematical objects in the brain and gets ready
for subsequent mathematical operation. The second is that, the mathematical
concept or operation can be understood or achieved by a simple operation on
the graphical objects. This is the step to be carried out in the brain, and
preferred to be simple (or at least conceptually simple). If a balance could
be achieved between these two ingredients in visualizing a mathematical
concept, then the graphical tool would be successful. This explains why the
Venn diagram has been so successful since it was introduced, and has now
become the standard graphical tool for set theory. Essentially, the Venn
diagram converts the set objects to graph objects in such a way that many set
relationships or operations could be accomplished by ‘reading’ the diagram—the
mathematical operation is done directly by the human visual system, instead of
having to invoke both the visual system and the brain. On the other hand, for
the tree diagram, each of the two ingredients does some job but there is room
for improvement.

The turtleback diagram we propose tries to optimize the two steps involved in
the design of a graphical tool for conditional probability. In particular, it
views events and the sample spaces as sets, and uses elements from Venn
diagrams—set intersection, complement and partition—for conditioning, with the
additional notion that the area of a set indicates probability whereas the
ratio of areas associated with relevant sets indicates conditional
probability. Once parts of the diagram are drawn and properly labelled, the
calculation of conditional probability involves just simple arithmetic on the
area of relevant sets. This makes it particularly easy to understand and use
for problem solving.

###  3.2 Other visual representations

There have been several prior attempts to represent conditional probability
visually [21, 9, 22, 23], and we discuss briefly three of these below.

####  3.2.1 Tomlinson-Quinn graphical model

This graphical model, for facilitating a visually moderated understanding of
conditional probability, described in [9], is a modified tree diagram.

Tomlinson and Quinn visualize compound events $A\cap B,A\cap\overline{B}$ as
nodes of a tree (see Figure 2 of [9], so essentially their idea is still a
tree diagram in which they carry out a Venn-diagram like visualization at each
tree node.

####  3.2.2 Roullete-wheel diagrams

Yamagishi [22] introduces roullete-wheel diagrams as a visual representation
tool; see Fig 1, p. 98 of [22]. He argues that

> “the graphical nature of [roulette-wheel diagrams] take advantage of
> people’s automatic visual computation in grasping the relationship between
> the prior and posterior probabilities.” (p. 105).

and provides experimental evidence that use of roulette-wheel diagrams
increases understanding of conditional probability beyond that for tree
diagrams. In this regard, Sloman et al. [24] state:

> “The studies reported support the nested-sets hypothesis over the natural
> frequency hypothesis. …The nested-sets hypothesis is the general claim that
> making nested-set relations transparent will increase the coherence of
> probability judgment.” (p. 307)

####  3.2.3 Iconic diagrams

“Iconicity” is the lowest of Terrence Deacon’s three levels of symbolic
interpretation111In Deacon’s framework, there are three levels of referential
relationship in a cognitive process, including iconic, indexical, and symbolic
reference, where higher levels are built hierarchically upon lower levels.
[25], as it is for Peirce on whose semiotic work Deacon’s theory is based. An
icon is a form of graphical representation that requires no significant depth
of interpretation: an icon brings to mind, without any apparent intermediate
thought, something that it resembles. For example, the diagram in Figure 2 is
universally iconic for human beings.

Figure 2: A diagram that is universally iconic for humans.

Brase [23] carried out a number of experiments from which he inferred that an
iconic representation of a Bayesian probability question is more effective in
eliciting correct responses than either no visual aids, or Venn diagrams.

A modified version of Brase’s question is as follows:

> “A new test has been developed for a particular form of cancer found only in
> women. This new test is not completely accurate. Data from other tests
> indicate a woman has 7 chances out of 100 of having cancer. The test
> indicated positively only 5 of these women as having cancer. On the other
> hand, the test indicated a positive result for 14 of the 93 women without
> cancer.
>
> Janine is tested for cancer with this new test. Janine has probability  of a
> positive result from the test, with a probability  of actually having
> cancer.”

An iconic representation for this problem is shown in Figure 3.

Figure 3: An iconic representation of the effectiveness of a cancer test.

The strength of such iconic representations is that they reduce the
calculation of probabilities to simple counting problems and, as Brase [23]
demonstrates, are effective in assisting students to get correct answers. A
weakness of iconic representations such as these, are that they rely on
counting discrete items and so are quite limited in representing more
realistic probabilities.

###  3.3 Turtleback diagrams

Our focus is on how to represent an event graphically, how to relate it to the
sample space, how to express the notion of conditioning such that it would be
easy to understand the concept of conditional probability, to gather pieces of
information together, and to solve problems accordingly.

We start by treating the sample space (denoted by $\Omega$) and events as
sets, and in terms of graph, as a region and its sub-regions, similarly as in
a Venn diagram. Assume the region representing the original sample space
$\Omega$ has an area of $1$. To simplify our discussion (or to abuse the
notation), we will use a label, say $``B"$, to denote the region associated
with event $B$. Note that here the label can be either a single letter, or
several letters (such a case indicates the intersection of events. For
example, a label $``AB"$ indicates the intersection of events $A$ and $B$ and
thus that of regions $A$ and $B$). Similarly we can use the union of two
regions (viewed as sets) to represent the union of two events. Other
operations of events can also be defined accordingly in terms of set
operations; we omit the details here. To quantify the chance of an event, we
associate it with the area of the relevant region. For example,
$\mathbb{P}(B)$ is indicated by the area of region $B$.

The centerpiece in ‘graphing’ conditional probability is to express the notion
of conditioning. This can be achieved by re-examining the definition of
conditional probability as given in (1). It can be interpreted as follows. Let
$A$ be the event of interest. Upon conditioning, say, on event $B$, both the
new effective sample space and event $A$ in this new sample space can be
viewed as their restriction on $B$, that is, $\Omega$ becomes $\Omega\cap B=B$
and $A$ becomes $A\cap B$, respectively. The conditional probability
$\mathbb{P}(A|B)$ can now be interpreted as the proportion of the part of $A$
that is inside $B$ (i.e., $A\cap B$) out of region $B$, that is,

$\mathbb{P}(A|B)=\frac{\mbox{area of region~{}}A\cap B}{\mbox{area of~{}}B}.$
(2)

Now we can describe how to sketch a turtleback diagram. We start by drawing a
circular disk which represents the sample space $\Omega$. Then we represent
events by partitioning the circular disk and the resulting subregions. To
facilitate our discussion, we define the partition of a set [26].
$\mathcal{P}=\\{S_{i}|~{}i\in\mathbb{I}\\}$ is a partition of set $S$ if
$S=\cup_{i}S_{i}$ and $S_{i}\subseteq S$, $S_{i}\cap S_{j}=\emptyset$ for all
$i\neq j\in\mathbb{I}$.

Figure 4: Illustration of the turtleback diagram for conditional probability.
The left panel shows the partition of $\Omega$ by $\Omega=B\cup\overline{B}$,
the middle panel shows event $B$ is further partitioned by $B=(A\cap
B)\cup(\overline{A}\cap B)$, and the right panel is a simplified version of
the middle panel where $``AB"$ stands for $A\cap B$, and $``\overline{A}B"$
stands for $\overline{A}\cap B$. The conditional probability $\mathbb{P}(A|B)$
is the ratio of the area for region “bcdb” and that for the area “abcda”.

We will use Figure 4 to assist our description. To represent the partition
$\Omega=\overline{B}\cup B$, we use a straight line “adc” to split the
circular disk into two halves, i.e., regions surrounded by “abcda” and
“adcea”, which stands for event $B$ and $\overline{B}$, respectively. The
regions corresponding to event $B$ and $\overline{B}$ can be further split for
a more refined representation involving other events. To represent conditional
probability as defined by (1), event $B$ is written as

$B=(A\cap B)\cup(\overline{A}\cap B),$ (3)

which can be represented by splitting the region for $B$, i.e., “abcda”, with
a straight line “db”. The conditional probability $\mathbb{P}(A|B)$ can then
be calculated as the ratio of the area for region “bcdb” and that for region
“abcda”.

The turtleback diagram leads to a partition of the sample space $\Omega$ as
follows

$\displaystyle\Omega$ $\displaystyle=$ $\displaystyle\overline{B}\cup B$ (4)
$\displaystyle=$ $\displaystyle\overline{B}\cup(A\cap B)\cup(\overline{A}\cap
B).$ (5)

Continuing this process, we can define events as complicated as we like in a
simple hierarchical fashion as a nesting sequence of partitions
$\mathcal{P}_{0}\succ\mathcal{P}_{1}\succ\mathcal{P}_{2}\succ...$ where
$\mathcal{P}_{0}=\\{\Omega\\}$, $\mathcal{P}_{1}=\\{B,\overline{B}\\}$, and
$\mathcal{P}_{i+1}$ is a refinement of $\mathcal{P}_{i}$ for index $i>0$ in
the sense that each element in $\mathcal{P}_{i+1}$ is a subset of some element
in $\mathcal{P}_{i}$.

We can now assign labels to each of the sub-regions, e.g., by the name of the
relevant events to indicate that a particular region is associated with that
event. For example in Figure 4, we assign labels $``AB"$ and
$``\overline{A}B"$ to regions “bcdb” and “abda”, respectively. Here, $``AB"$
means $A\cap B$, and $``\overline{A}B"$ indicates $\overline{A}\cap B$, and
the same convention carries over throughout. Accordingly, the turtleback
diagram simplifies to the right panel in Figure 4. Note that here an event
need not be a connected region, rather it could be a collection of patches
(i.e., small regions) with each of them capturing information from a different
source. This causes a little burden in calculation but costs really nothing
conceptually, or, in terms of the ability of visualization.

One advantage of such a recursive-partition representation of the sample space
$\Omega$ is that the data are now highly organized and we can easily operate
on it, for example to find out the probability of a certain event. The idea of
organizing the data via recursive space-partition and manipulating by their
labels has been explored in CART (classification and regression trees [27])
and more recently, random projection trees [28], as well as a recent work of
one author and his colleagues [29]. Note that dividing a region into a number
of small patches also entails the total probability formula, an important
ingredient in conditional probability to which formula (3) is related. We will
use the ‘Lung disease and smoking’ example to illustrate the use of turtleback
diagrams for conditional probability.

Figure 5: The turtleback diagram for the ‘Lung disease and smoking’ example.
The letters $``L"$ and $``\overline{L}"$ stand for “with lung disease” and
“without lung disease”, $``S"$ and “$\overline{S}$” for “smoking” and
“nonsmoking”, respectively.

###  3.4 The lung disease and smoking example

This example is taken from online sources (see [30]). It is described as
follows.

> “According to the Arizona Chapter of the American Lung Association, $6.0\%$
> of population have lung disease. Of those having lung disease, $92.0\%$ are
> smokers; of those not having lung disease, only $24.0\%$ are smokers. Answer
> the following questions.
>
>   1. (1)
>
> If a person is randomly selected in the population, what is the chance that
> she is a smoker having lung disease?
>
>   2. (2)
>
> If a person is randomly selected in the population, what is the chance that
> she is a smoker?
>
>   3. (3)
>
> If a person is randomly selected and is discovered to be a smoker, what is
> the chance that she has lung disease?”
>
>

According to the information given in the problem, we can sketch a graph as
Figure 5. Labels and area information to each sub-regions are assigned
properly. Assume the circular disk has an area of $1$. Now we can answer the
questions quickly as follows.

  1. (1)

The answer is simply the area of region “adba”, which is $6\%\cdot
92\%=0.0552$.

  2. (2)

The answer is the area of region “edbae”, which is $6\%\cdot 92\%+94\%\cdot
24\%=0.2808$. This is, in essence, the total probability formula
$\mathbb{P}(S)=\mathbb{P}(L\cap S)+\mathbb{P}(\overline{L}\cap S)$.

  3. (3)

Recognizing that this involves conditional probability and is the ratio of two
relevant areas,
(area of “adba”/area of “edbae”)=0.0552/0.2808=0.1966.

###  3.5 Difficulty with the Venn diagram

The Venn diagram is known as the standard graphical tool for set theory. Both
Venn diagram and the turtleback diagram use regions to represent sets.
However, there is a major difference. In a turtleback diagram, as illustrated
in Figure 4, straight lines, such as line “adc”, “db” etc, are used to split
the sample space and regions. In contrast, the Venn diagram represents events
by drawing circular disks. Partitioning the sample space $\Omega$ in such a
way would cause substantial difficulty in handling the complement operation,
one crucial ingredient in conditional probability. One has to deal with a
setting where the complement of a region would surround the region itself, for
example, in Figure 6, $``\overline{S}"$ and $``\overline{L}"$ surround $``S"$
and $``L"$, respectively. This would cause extra burden to the human brain or
the visual system. We will illustrate with the ‘Lung disease and smoking’
example.

Figure 6: The Venn diagram approach to the ‘Lung disease and smoking’ example.

In Figure 6, one would find it tricky to label the region and put area
information for $``\overline{L}"$ (which is $94\%$) without causing confusion.
Moreover, it may require some extra work (versus simply “reading” from the
graph) to assign the label $``\overline{L}\overline{S}"$, or to calculate the
area of this region. In contrast, the turtleback diagram (c.f. Figure 5)
introduces straight lines, e.g., “adc”, “ed”, and “db”, which readily avoids
obstacles caused by set intersections or complements in a Venn diagram.

##  4 Semantic equivalence of the turtleback and the tree diagram

Given a graphical representation, it is natural to ask questions about its
expressive power–will it be expressive enough to represent a complicated or
very abstract concept? We will show that the turtleback diagram is equally
expressive as the tree diagram.

The way that the turtleback diagram progressively refines the partition over
the sample space is essentially a recursive space partition, where the sets
involved in the partition are organized as a chain of enclosing sets. For
example, in Figure 4, we have

$\left(A\cap B\right)\subseteq
B\subseteq\Omega,~{}\mbox{and}~{}\left(\overline{A}\cap B\right)\subseteq
B\subseteq\Omega.$ (6)

By equivalence (see, for example, [27]) between the recursive space partition
and the tree structure, we can actually show the “semantic” equivalence
between the turtleback diagram and the tree diagram. The remaining of this
section is dedicated to this. Let a tree node correspond to a set in a
recursive space partition with the following three properties:

  * 1)

The root node corresponds to the sample space $\Omega$;

  * 2)

All the child nodes of a node form a decomposition of this node;

  * 3)

Down from the root node, the nodes along any path form a chain of enclosing
sets.

Property 2) entails the total probability formula, and property 3) corresponds
to a refinement of a partition. This allows one to turn the turtleback diagram
in Figure 4 into a tree representation, that is, the left panel of Figure 7.
The “chain” property forces a child node to be a restriction of its parent
node. We can use this to simplify the labels for the tree nodes, e.g., the
left panel becomes the right in Figure 7. Note that in the right panel, really
node $A$ corresponds to the set $\Omega\cap B\cap A$, that is, the
intersection of all sets along the path from the root to node $A$ (i.e., the
tree path $*\rightarrow B\rightarrow A$).

Figure 7: The tree diagram representation of the turtleback diagram in Figure
4.

For real world conditional probability problems, often the following formula
is used instead of (1), due to availability of information from multiple
sources

$\mathbb{P}(A|B)=\frac{\mathbb{P}(A\cap B)}{\sum_{i}\mathbb{P}(B\cap A_{i})}$
(7)

where $\sum_{i}A_{i}=\Omega$. This requires the calculation of probabilities
in the form of $\mathbb{P}(B\cap A_{i})$, or in other words, the probability
of the intersection of multiple events.

In Figure 7, by construction node $A$, through path $*\rightarrow B\rightarrow
A$, has a size $\mathbb{P}(\Omega\cap B\cap A)$, and node $B$ has size
$\mathbb{P}(B)$. We can now endow the weight of edge $B\rightarrow A$
according to the proportion of node $A$ (treated as a subset of $B$) out of
$B$, or the probability of transition to node $A$ given that one has reached
node $B$ from the root. This equals $\mathbb{P}(A|B)$. Such a definition is
valid as the size of nodes $A,\overline{A}$ and $B$ satisfies
$\mathbb{P}(A\cap B)+\mathbb{P}(\overline{A}\cap B)=\mathbb{P}(B)$.

Figure 8: The tree diagram approach illustrated.

Thus, in Figure 8, the probability that one arrives at a node, say $A$, along
the path $\Omega\rightarrow B\rightarrow A$ is given by

$\mathbb{P}(\Omega BA)=\mathbb{P}(\Omega\cap B\cap
A)=\mathbb{P}(BA)=\mathbb{P}(B)\cdot\mathbb{P}(A|B),$ (8)

which is simply the product of edge weights along the path $\Omega\rightarrow
B\rightarrow A$ (the edge weight for $\Omega\rightarrow B$ is
$\mathbb{P}(B)$). Same reasoning extends to any node in a tree. Thus we have
provided a tree-based interpretation of the turtleback diagram for conditional
probability. Such an algebraic system on the tree has the following two
properties:

  1. 1.

The probability of arriving at any node equals the product of edge weights
along the path.

  2. 2.

The weight of an edge $H\rightarrow L$ has weight given by
$\mathbb{P}(L|*,...,H)$.

This is exactly what a tree diagram would represent. The above properties
extend readily to a series of events. For example, the probability of a series
of events, $B\rightarrow C\rightarrow D$ can be computed as the probability of
arriving at node $D$ along the tree path $\star\rightarrow B\rightarrow
C\rightarrow D$ (c.f., Figure 8)

$\displaystyle\mathbb{P}(B\cap C\cap D)$ $\displaystyle=$
$\displaystyle\mathbb{P}(*\rightarrow B)\cdot\mathbb{P}(B\rightarrow
C)\cdot\mathbb{P}(C\rightarrow D)$ (9) $\displaystyle=$
$\displaystyle\mathbb{P}(B)\cdot\mathbb{P}(C|B)\cdot\mathbb{P}(D|B,C).$

This approach applies even for non-sequential events, as one can artificially
attach an order to the events according to the “arrival” of relevant
information. Thus, we have shown the semantic equivalence between the
turtleback diagram and the tree diagram. Their difference is mainly on the
visual representation, which matters as visual tools.

The tree diagram appears to be less intuitive than the turtleback diagram as
there is no longer an association between the area of a region and its
probability (one may use the thickness of an edge to indicate the probability,
but that is less attractive too). However, the tree diagram seems to scale
better to large problems.

##  5 Case studies

We consider four examples in case study, including the ‘Lung disease and
smoking’ example, the ‘History and war’ example, the ‘Lucky draw’ example, and
‘the urn model’ example [1]. As a matter of fact, very few students (about
$10-15\%$) can do the ‘History and war’ example completely correctly in an in-
class practice, after explaining to them the non-graph based concept of
conditional probability. That motivated us to adopt the graph-based approach.
In the following, we provide the details of the examples.

###  5.1 The lung disease and smoking example

Figure 9: The tree diagram approach for the ‘Lung disease and smoking’
example. The letters $``L"$ and $``\overline{L}"$ stand for “with lung
disease” and “without lung disease”, $``S"$ and $``\overline{S}"$ for
“smoking” and “nonsmoking”, respectively.

With the tree diagram, the answer to (1) is the probability of reaching node
$S$ along the path $\star\rightarrow L\rightarrow S$, which is the product of
edge weights along this path and is calculated as $6\%\cdot 92\%=0.0552$. The
solution to (2) is the sum of products of edge weights along two paths,
$\star\rightarrow L\rightarrow S$ and $\star\rightarrow\overline{L}\rightarrow
S$, that is, $6\%\cdot 92\%+94\%\cdot 24\%=0.2808$, and (3) by the ratio of
the product of edge weights along path $\star\rightarrow L\rightarrow$ over
that over two paths, which is $0.0552/0.2808=0.1966$.

###  5.2 The History and War example

This example is artificially created so that it has a similar problem
structure as the ‘Lung disease and smoking’ example. It is described as
follows.

> “According to a market research about the preference of movies, $10\%$ of
> the population like movies related to history. Of those who like movies
> related to history, $90\%$ also like movies related to wars; of those who do
> not like movies related to history, only $30\%$ like movies related to wars.
> Answer the following questions.
>
>   1. (a)
>
> If a person is randomly selected in the population, what is the chance that
> she likes both movies related to wars and movies related to history?
>
>   2. (b)
>
> If a person is randomly selected in the population, what is the chance that
> she likes movies related to wars?
>
>   3. (c)
>
> If a person is randomly selected and is discovered to like movies related to
> wars, what is the chance that she likes movies related to history?”
>
>

We can construct a turtleback diagram as the left panel of Figure 10. One can
quickly answer the questions as follows. (a) is the area of region “adba”,
which is given by $10\%\cdot 90\%=0.09$, (b) is the total area of region
“edbae”, which is given by $10\%\cdot 90\%+90\%\cdot 30\%=0.36$, and (c) is
the ratio of (a) and (b) which is $0.09/0.36=0.25$.

Figure 10: Solving the ‘History and War’ example with the turtleback diagram
and tree diagram, respectively. The letters $``H"$ and $``\overline{H}"$ stand
for “like movies related to history” and “do not like movies related to
history”, $``W"$ and $``\overline{W}"$ for “like movies related to wars” and
“do not like movies related to wars”, respectively.

Similarly, the right panel of Figure 10 is a tree diagram. One can answer the
questions as follows. (a) is the product of edge weights along the path
$\star\rightarrow H\rightarrow W$, which is given by $10\%\cdot 90\%=0.09$,
(b) is the sum of the product of edge weights along two paths,
$\star\rightarrow H\rightarrow W$ and $\star\rightarrow\overline{H}\rightarrow
W$, which is given by $10\%\cdot 90\%+90\%\cdot 30\%=0.36$, and (c) is the
ratio of (a) and (b) which is $0.09/0.36=0.25$.

###  5.3 The lucky draw example

The lucky draw example is taken from the popular lucky draw game. This example
is especially useful as many sampling without replacement problems can be
converted to this and solved easily. Here we take a simplified version with
the total number of tickets being $5$ and there is only one prize ticket. The
description is as follows.

> “There are $5$ tickets in a box with one being the prize ticket. $5$ people
> each randomly draws one ticket from the box without returning the drawn
> ticket to the box. Is this a fair game (i.e., each draws the prize ticket
> with the same chance)?”

Figure 11: The tree diagram for the ‘Luck draw’ game. The letters $``P"$ and
$``N"$ denote the prize ticket and non-prize ticket, respectively.

Figure 11 depicts the process of ticket drawing. As here our interest is the
prize ticket, the tree branch that has already seen the prize ticket will not
grow further. Easily the probability of getting the prize ticket at the first
draw is $1/5$. Following Figure 11, the probability of getting the prize
ticket at the second draw is the product of edge weights along the path
$\star\rightarrow N\rightarrow P$, which is $0.8\cdot 0.25=0.2$. Similarly,
the probability of getting the prize ticket at the third draw is given by
$0.8\cdot 0.75\cdot 1/3=0.2$, and so on.

Figure 12: The turtleback diagram for the ‘Luck draw’ game. The letters in the
labels indicates status of each attempt, $``P"$ for prize and $``N"$ for a
non-prize ticket. For example, “NNP” means getting non-prize tickets for the
first two draws and the prize ticket at the third draw. The percentage next to
the label indicates the probability of a prize at the last draw, conditional
on the outcome of all preceding draws. For example, “25%” next to “NP” means
the conditional probability of getting a prize is 25% in the second draw if
the first draw is not a prize. Or in other words, that is the ratio of the
area of the slice containing “NP” to all slices after the first slice is taken
away.

Figure 12 is the turtleback diagram for the ‘Luck draw’ game. Easily the
probability of getting the prize ticket at the first draw is the area of the
region labelled as “P”, which is $0.2$. Following the figure, the probability
of getting the prize ticket at the second draw is the area of the region
labelled as “NP”, which is $0.8\cdot 25\%=0.2$. Similarly, the probability of
getting the prize ticket at the third draw is given by $0.8\cdot 75\%\cdot
1/3=0.2$, and so on.

###  5.4 An urn model example

This can be viewed as an extension of the lucky draw problem in the sense that
there are more than one prize tickets here. Note that this example mainly
serves to demonstrate that both the tree and the turtleback diagram could be
used to solve problems of such a complexity (one can solve this problem
quickly by distinguishing the two green balls and apply result of the lucky
draw game222Label the two green balls as $G_{1}$ and $G_{2}$, respectively.
Then the probability of getting a green ball at each draw is simply that of
getting $G_{1}$ or $G_{2}$. Either $G_{1}$ or $G_{2}$ can be treated as the
only prize ticket in the lucky draw game thus the probability of getting
either one is $1/5$, and so the probability of getting a green ball at any
draw is always $2/5$.). Assume there are $2$ greens balls and $3$ red balls.
The problem is described as follows.

> “There are $2$ green balls and $3$ red balls in an urn. One randomly picks
> one ball for five times from the urn without returning. Will each draw have
> the same chance of getting the green ball?”

Figure 13: The tree-based approach for an urn model with $2$ green balls and
$3$ red balls. We use the $\star$ to denote the root node, and “R” and “G” for
red ball and green ball, respectively. As our interest is the green balls, so
tree branches that have seen $2$ green balls will not grow any further. Figure
14: The turtleback diagram for an urn model with $2$ green balls and $3$ red
balls. “R” and “G” are used to denote red ball and green ball, respectively.
As our interest is the green balls, so tree branches that have seen $2$ green
balls will not grow any further. Each letter, “G” or “R”, indicates the
outcome of a particular draw. For example, a “RGRG” indicates that the first
draw gets a red ball, the second draw a green ball, the third a red and the
fourth a green ball.

Figure 13 is the tree diagram for the urn model. We are not going to calculate
the probability of getting a green ball for each draw, instead we only do it
for the third draw. The probability of getting a green ball at the third draw
is give by the sum of the product of edge weights along three paths

$\star\rightarrow G\rightarrow R\rightarrow G,~{}\star\rightarrow R\rightarrow
R\rightarrow G,~{}\star\rightarrow R\rightarrow G\rightarrow G,$

which is $(2/5)(3/4)(1/3)+(3/5)(1/2)(2/3)+(3/5)(1/2)(1/3)=2/5$. One can
similarly calculate that the probability of getting a green ball at other
draws all equal to $2/5$.

Figure 14 is the turtleback diagram for the urn model. To calculate the
probability that the third draw gets a green ball, we simply sum up the area
of all regions with a label such that the third letter is “G”. That is, the
total area of regions labelled as

$``RGG",~{}``GRG",~{}``RRGG",~{}``RRGRG",$

which is

$\frac{3}{5}\cdot\frac{1}{2}\cdot\frac{1}{3}+\frac{2}{5}\cdot\frac{3}{4}\cdot\frac{1}{3}+\frac{3}{5}\cdot\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{1}{2}+\frac{3}{5}\cdot\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{1}{2}=0.4.$

The calculation seems a little tedious, but conceptually very simple, as long
as one could follow the way the regions are partitioned.

Course | # students | Class size | Institute
---|---|---|---
STAT235 | 128 | 40-60 | UMKC
MTH231 | 25 | 10-20 | UMassD
MTH331 | 72 | 35-45 | UMassD
Table 1: Students involved in the case studies.

##  6 Empirical data

We carried out case studies on over $200$ students. This includes students in
the elementary statistics class, STAT235 (non-calculus based), at University
of Missouri Kansas City (UMKC) during 2012-2013, and students from elementary
statistics, MTH231, and elementary probability, MTH331, classes at University
of Massachusetts Dartmouth (UMassD) during 2015-2017. These three courses had
a fairly different student population. For STAT235, about 30% from
engineering, 30% from business, and the rest from such diverse majors as
biology, chemistry, psychology, political sciences, education etc. For MTH231,
about 80% are from mathematics or data science, and the rest from majors such
as computer science, electrical engineering, criminal justice etc. For MTH331,
about 75% from computer science or electrical engineering, 20% from
mathematics or data science, and the rest from other engineering majors or
economics, physics etc. Table 1 gives a summary of students involved in the
case studies.

The study is carried out as follows. First we explain to students the concept
of conditional probability with a non-graph based approach. Then we continue
with two exercises. In the first exercise, we explain to students the ‘Lung
disease and smoking’ example, with both the turtleback and the tree diagram,
and have students solve the ‘History and war’ problem, or vice versa (for
different classes we were teaching). In another exercise, we explain the
‘Lucky draw’ example, and have students solve the ‘Urn model’ problem, or vice
versa. Due to time constraints on the course schedule, we did not ask students
to solve problems using a particular technique followed by its discussion.
Rather we discussed both the turtleback and the tree diagrams, and let
students choose one of them for problem solving. Table 2 is a breakdown of the
number of students involved.

Course | Lung disease and | War and history | Lucky | Urn
---|---|---|---|---
| smoking | movie | draw | model
STAT235 | 66 | 62 | 62 | 66
MTH231 | 14 | 11 | 14 | 11
MTH331 | 37 | 35 | 35 | 37
Total | 117 | 108 | 111 | 114
Table 2: Number of students involved in the empirical study breakdown by course and problem. | Neither | Either one | Prefer | Prefer
---|---|---|---|---
Question | helpful | helpful | Turtleback | Tree
Lung disease and smoking | 13.7% | 86.3% | 53.0% | 33.3%
War and history movies | 11.1% | 88.9% | 54.6% | 34.3%
Lucky draw | 17.1% | 82.9% | 34.2% | 48.6%
Urn model | 21.9% | 78.1% | 31.6% | 46.5%
Table 3: Data collected in the case studies on whether graphs help understand
the concept of conditional probability, and the preference between the
turtleback diagram and the tree diagram.

We collect two types of data from the case studies, one on students’
preference between graph and non-graph based approach, and the other on
students’ preference between the turtleback and the tree diagram. Here, except
for the case of non-graph based approach, by preference we mean the students
actually used the technique for problem solving, and nearly in all such cases
they could apply it correctly in solving the assigned problem; so we use this
as measurement of learning outcome (with an understanding that further
experiments may be needed to validate this). The results are reported in Table
3. The data collected are quite encouraging. About 78-88% students found a
graph tool helpful. For the ‘Lucky draw’ and the ‘Urn model’, fewer students
found it helpful. This is possibly because these two problems appear to be
harder to students: even a graphical tool may not help them much. Further
experiments are needed to validate or understand this.

In terms of a preference for which graphical tool, the results show an
interesting pattern. For the ‘Lung disease and smoking’ and the ‘War and
history’ example, more students prefer the turtleback diagram to the tree
diagram, around 53-54% vs 33-34%. The ‘Lucky draw’ and the ‘Urn model’
examples exhibit an opposite pattern, more students prefer the tree diagram to
the turtleback diagram, around 46-48% vs 31-34%333Since in all cases, the
sample size is large enough and the difference between contrast groups is
significant, we did not carry out a hypothesis testing using the reported
data.. This is probably due to the fact that, in the first two examples, the
sample spaces and events involve populations in the usual sense, while the
last two examples involve sequential decisions, for which a tree structure
that represents the decision dichotomy may be more natural (although in such
cases, the concept of conditional probability is not as natural as that in the
turtleback diagram). Further experiments are needed to confirm this. The
advantage of the turtleback diagram over the tree diagram appears to decrease
as the problem becomes harder, but this is not a serious problem for beginning
students as those who most need help from a graphical representation are just
those who could not solve simple problems. Moreover, we do not expect one
single graphical tool can help solve all the problems, rather different people
may use different tools for a particular problem.

##  7 Potential research questions

Many instances of conditional probability occur in sampling without
replacement. Tarr and Jones [8] describe a framework for assessing middle
school students’ thinking in conditional probability and independence, which
is elaborated in [12]. This framework is a levels model, with 4
levels—Subjective, Transitional, Informal Quantitative, and Numerical—subject
to all the difficulties such a model has as students transition from one level
to another.

Research Question 1: Are turtleback diagrams, as compared to tree diagrams,
helpful to students, at any or all of the Tarr-Jones framework levels, in
understanding conditional probability. If so, how can we measure and assess
the comparative utility of turtleback diagrams compared to tree diagrams?

Research Question 2: Related to Research Question 1, specifically, how helpful
are turtleback diagrams in helping students understand conditional probability
in the context of sampling without replacement?

Conditional probability is increasingly being introduced into middle school in
the United States. The Conference Board of the Mathematical Sciences [31]
stated:

> Of all the mathematical topics now appearing in middle grades curricula,
> teachers are least prepared to teach statistics and probability. Many
> prospective teachers have not encountered the fundamental ideas of modern
> statistics in their own K-12 mathematics courses…Even those who have had a
> statistics course probably have not seen material appropriate for inclusion
> in middle grades curricula. (p. 114)

Research Question 3: Are turtleback diagrams helpful to middle school teachers
of probability and statistics in (a) enhancing their own understanding of
conditional probability and (b) assisting them to better teach conditional
probability? If so, how and to what extent?

##  8 Conclusions

Motivated by difficulties encountered by many undergraduate students new to
statistics, we re-examined the definition and representation of conditional
probability, and presented a Venn-diagram like approach: the turtleback
diagram. We discussed our graphical tool in the context of other graphical
models for conditional probability, and carried out case studies on over $200$
students of elementary statistics or probability classes. Our case study
results are encouraging and the graph-based approaches could potentially lead
to significant improvements in both the students’ understanding of conditional
probability and problem solving. While the existing tree diagram is preferred
to the turtleback diagram on problems that involve a sequential decision, the
turtleback diagram is considered more helpful in settings where the underlying
population resembles the usual human population; it is exactly in such
situations that weaker students are more likely to need help. Though the
turtleback diagram appears very different from the tree diagram, we are able
to unify them and show their equivalence in terms of semantics.

Our discussion suggests a simple framework for visualizing abstract concepts,
that is, a suitable graph representation of the abstract concept followed by a
simple post-processing in the visual-brain system. A good visualization idea
needs to balance both. We are able to use such a framework to interpret the
difficulty encountered by the tree diagram, and aid our development of the
turtleback diagram. Further studies are expected to validate or to adopt such
a framework to general visualization tasks. Given the increasingly important
role played by data visualization in data science and exploratory data
analysis [15, 32, 16, 33], it would be worthwhile to give a few remarks here
comparing the graph representation of abstract concepts and data
visualization. These two concepts are different yet closely related. Graphical
representation aims to understand an abstract (or complicated) concept by
representing elements of the concept with a graph, while data visualization
seeks to understand the data or the information behind by displaying aspects
(i.e., descriptive statistics) of the data. In terms of implementation, as
both aim to help understanding or reasoning, the used graphical objects need
to be simple (though simple in different ways in the two cases). In data
visualization, the graphical objects need to be simple so that people can
quickly grasp the information conveyed or to understand the concept behind
without resorting to paper and pencil; in graphical representation of
concepts, the objects need to be conceptually simple and easy to manipulate
for applications of the concepts.

Our case studies suggest that it is worthwhile to introduce such graphical
tools to students whose success would seem to depend on them. We hope that
this will benefit our statistics colleagues who are teaching elementary
statistics and students who are struggling with the concept of conditional
probability and its application to problem solving. The potential savings in
time can be huge. As a conservative estimate, assume each year there are about
$1.5$ million bachelor’s degrees awarded in US (about $1.67$ million awarded
in 2009). Assume there are about $200,000$ of them have taken an elementary
statistics class, and about $10\%$ of them need help and succeed with our
proposed approach, and further assume an average class size of $40$. If each
instructor saves $2$ hours of time in each elementary statistics class and
each student who benefits from our approach saves $1$ hour, then the estimated
total amount of time saved is at least $30,000$ hours per year in the U.S.
alone.

## Acknowledgement

The authors are grateful to Professor Yong Zeng at UMKC for kindly pointing to
the ‘Lung disease and smoking’ example, and for encouragement and support on
some of the case studies.

## References

  * [1] Rice, J. (1995) Mathematical Statistics and Data Analysis (Second Edition). Duxbury Press.
  * [2] Johnson, R. and Tsui, K. (2003) Statistical Reasoning and Methods. John Wiley.
  * [3] Ancker, J. S. (2006) The language of conditional probability. Journal of Statistics Education, 14, 1-5.
  * [4] Mann, P. (2003) Introductory Statistics. John Wiley.
  * [5] Tversky, A. and Kahneman, D. (1980) Causal schemas in judgments under uncertainty. Progress in social psychology, 1, 49-72.
  * [6] Fischbein, E. and Gazit, A. (1984) Does the teaching of probability improve probabilistic intuitions? Educational studies in mathematics, 15, 1-24.
  * [7] Falk, R. (1986) Conditional probabilities: insights and difficulties. Proceedings of the Second International Conference on Teaching Statistics, 292-297.
  * [8] Tarr, J. E. and Jones, G. A. (1997) A framework for assessing middle school students? Thinking in conditional probability and independence. Mathematics Education Research Journal, 9, 39-59.
  * [9] Tomlinson, S. and Quinn, R. (1997) Understanding conditional probability. Teaching Statistics, 19, 2-7.
  * [10] Tarr, J. E. (2002) The confounding effects of the phrase ’50-50 Chance’ in making conditional probability judgments. Focus on Learning Problems in Mathematics, 24, 35-53.
  * [11] Yáñez, G. C. (2002) Some challenges for the use of computer simulations for solving conditional probability problems. The 6th International Conference on Teaching Statistics, Cape Town, South Africa.
  * [12] Tarr, J. E. and Lannin, J. K. (2005) How can teachers build notions of conditional probability and independence? In Exploring Probability in School (pp. 215-238), Springer, US.
  * [13] Arcavi, A. (2003) The role of visual representations in the learning of mathematics. Educational studies in mathematics, 52, 215-241.
  * [14] Presmeg, N. C. (2006) Research on visualization in learning and teaching mathematics. Handbook of research on the psychology of mathematics education, Sense Publishers, Rotterdam, 205-235.
  * [15] Tukey J. (1977) Exploratory Data Analysis. Addison-Wesley.
  * [16] Cleveland, W. S. (1993) Visualizing Data. Hobart Press.
  * [17] Morris, D. (2016) Bayes’ Theorem Examples: A Visual Introduction For Beginners. ISBN-13: 978-1549761744.
  * [18] Collins, R. (2017) Bayes Theorem Examples: Visual Book for Beginners. CreateSpace Independent Publishing Platform ISBN-13: 978-1547270385.
  * [19] Gelman, A. and Carlin, J. B. and Stern, H.S. and Dunson, D.B. and Vehtari, A. and Rubin, D.B. (2013) Bayesian Data Analysis, 3rd Edition, Chapman and Hall, London.
  * [20] Edwards, A. W.F. (2004) Cogwheels of the Mind: The Story of Venn Diagrams. Johns Hopkins University Press, Baltimore, MD.
  * [21] Gigerenzer, G. and Hoffrage, U. (1995) How to improve Bayesian reasoning without instruction: frequency formats. Psychological review, 102(4), 684-705.
  * [22] Yamagishi, K. (2003) Facilitating normative judgments of conditional probability: Frequency or nested sets? Experimental Psychology, 50, 97-106.
  * [23] Brase, G. L. (2009) Pictorial representations in statistical reasoning. Applied Cognitive Psychology, 23, 369-381.
  * [24] Sloman, S. A., Over, D., Slovak, L. and Stibel, J. M. (2003) Frequency illusions and other fallacies. Organizational Behavior and Human Decision Processes, 91, 296-309.
  * [25] Deacon, T. W. (1998) The symbolic species: The co-evolution of language and the brain. WW Norton & Company, NY.
  * [26] Chartrand, G., Zhang, P. (2011) Discrete Mathematics. Waveland Press, Inc.
  * [27] Breiman, L., Friedman, J., Olshen, R. and Stone, C. (1984) Classification and Regression Trees. Wadsworth, CA.
  * [28] Dasgupta, S. and Freund, Y. (2008) Random projection trees and low dimensional manifolds. Fortieth ACM Symposium on Theory of Computing (STOC), 537-546.
  * [29] Yan, D., Huang, L. and Jordan, M. (2009) Fast approximate spectral clustering. Proceedings of the 15th international conference on Knowledge discovery and data mining (SIGKDD), 907-916.
  * [30] Weiss, N. A. (2012) Introductory Statistics. Addison-Wesley, 193.
  * [31] Conference Board of the Mathematical Sciences (2001) The mathematical education of teachers. American Mathematical Society, Providence, RI.
  * [32] Tufte E. (1983) The Visual Display of Quantitative Information. Graphics Press, CT.
  * [33] Yau, N. (2011) Visualize this: the flowing data guide to design, visualization, and statistics. Wiley.