Chapter 2 Elements of Set Theory for Probability
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Figure 2.1: The beaverduck from Tenso Graphics
In order to formalise the theory of probability, our first step consists in formulating events with uncertain outcomes as mathematical objects. To do this, we take advantage of Set Theory.
In this chapter, we will overview some elements of Set Theory, and link them with uncertain outcomes.
2.1 Random Experiments, Events and Sample Spaces
Let us start our journey with some definitions:
We perform Random Experiments almost every day in our everyday life. Some are very deliberate, such as the referee flipping a coin to decide who gets to kick the ball first in a football game. However, some are less deliberate, such as the rise or fall of the price of a Stock which, as we have illustrated in the introductory lecture, falls within this definition.
For instance, in some card games, you have to draw more than one card, and in some dice games, (such as the Bolivian game of âCacho Alalayâ), you have to throw more 5 dice. The result of your throw is therefore the composition of elementary events (the result of each die).
With a game of cards, we can also illustrate Composite Events, e.g. drawing 4 Aces
Figure 2.2: Someone is lucky!
2.2 Some definitions from set theory
The following definitions from set theory will be useful to deal with events.
2.3 The Venn diagram
The Venn diagram is an elementary schematic representation of sets and helps displaying their properties. As you might remember from your elementary mathematics classes, a Venn diagram represents a set with a closed figure and it is assumed that the set elements are in the surface contained by the figure. They are very useful to illustrate inclusion and equality as well as more abstract notions.
Figure 2.3: Inclusion and Equality of sets with Venn Diagrams
2.3.1 Sample Space and Events
By definition, an event or several events, constitute a subset of the sample space. This can be represented in a Venn Diagram with a figure enclosed within the Sample Space.
Figure 2.4: Venn Diagram - An event within the Sample Space
2.3.2 Exclusive and Non-Exclusive Events
Two events are mutually exclusive is they cannot occur jointly. This is represented by two separate enclosed surfaces within the sample space.
Figure 2.5: Venn Diagram - Two mutually exclusive events
Events that are not mutually exclusive have a shared area, which in set theory constitutes an intersection and gets shown in a Venn diagram by two figures that share some of their areas.
Figure 2.6: Venn Diagram - Two Non-mutually exclusive events
2.3.3 Union and Intersection of Events
The union of the events \(A\) and \(B\) is the event which occurs when either \(A\) or \(B\) occurs: \(A \cup B\). In a Venn diagram, we can illustrate this by shading the area enclosed by both sets.
Figure 2.7: Venn Diagram - \(A\cup B\)
The intersection of the events \(A\) and \(B\) is the event which occurs when both \(A\) and \(B\) occur: \(A \cap B\). In a Venn diagram, we can illustrate this by shading the area shared by both sets.
Figure 2.8: Venn Diagram - \(A\cap B\)
2.3.4 Complement
The complement of an event \(A\) is the event that occurs when \(A\) does not occur: We write \(A^{c}\) or \(\overline{A}\). A Venn diagram illustrates this by shading the area outside the set that defines the event \(A\).
Figure 2.9: Venn Diagram - \(A^{c}\)
Let \(S\) be the complete set of all possible events, i.e. the Sample Space. In such case, \(A^c\) can be written as: \[A^c = S \setminus A = S-A.\] and is such that \[A \cup A^c = S.\]
2.3.5 Some Properties of union and intersection
Let \(A\), \(B\), and \(D\) be sets. The following laws hold:
Commutative laws: Union and Intersection of sets are commutative, i.e. they produce the same outcome irrespective of the order in which the sets are written. \[\begin{eqnarray*} A \cup B = B \cup A \\ A \cap B = B \cap A \end{eqnarray*}\]
Associative laws: Union and Intersection of more than two sets operate irrespective of the order. \[\begin{eqnarray*} A \cup (B \cup C) = (A \cup C) \cup C \\ A \cap (B \cap C) = (A \cap C) \cap C \end{eqnarray*}\]
Distributive laws
- The intersection is distributive with respect to the union, i.e. the intersection between a set (\(A\)) and the union of two other sets (\(B\) and \(C\)) is the union of the intersections.
\[ A \cap (B \cup C) = (A \cap B) \cup (A\cap C)\]
- The union is distributive with respect to the intersection, i.e. the union between a set (\(A\)) and an intersection of two others is the intersection of the unions.
\[A \cup (B \cap C) = (A\cup C) \cap (A \cup C)\]
- The intersection is distributive with respect to the union, i.e. the intersection between a set (\(A\)) and the union of two other sets (\(B\) and \(C\)) is the union of the intersections.
\[ A \cap (B \cup C) = (A \cap B) \cup (A\cap C)\]
2.4 Countable and Uncountable sets
Events can be represented by means of sets and sets can be either countable or uncountable.
In mathematics, a Countable Set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers \(\mathbb{N}= \{0, 1, 2, 3, \dots \}\).
A countable set can be countably finite or countably infinite. Whether finite or infinite, the elements of a countable set can always be counted one at a time and, although the counting may never finish, every element of the set is associated with a natural number. Roughly speaking one can count the elements of the set using \(1,2,3,..\)
The identities in proposition ?? are helpful to define some other relations between sets/events.
2.5 De Morganâs Laws
Operations of union and intersection of sets have interesting interactions with complementarity. These properties are formulated under the name of De Morganâs laws.
For simplicity, let us illustrate these laws in the case of two sets \(A\) and \(B\) both included in the Sample Space \(S\). Just keep in mind that they can be generalized for an infinite amount of sets.
2.5.1 First Law
Let \(A\) and \(B\) be two sets in \(S\), then \[\begin{eqnarray} (A\cap B)^{c} =A^c \cup B^c, \end{eqnarray}\]
where:
- Left hand side: \((A\cap B)^{c}\) represents the set of all elements that are not both \(A\) and \(B\);
- Right hand side: \(A^c \cup B^c\) represents all elements that are not \(A\) (namely they are \(A^c\)) and not \(B\) either (namely they are \(B^c\)) \(\Rightarrow\) set of all elements that are not both \(A\) and \(B\).
Put in words, the firs law says that
the complement of the intersection between two sets is the union of their complements.
2.5.2 Second Law
Let \(A\) and \(B\) be two sets in \(S\). Then:
\[\begin{eqnarray} (A\cup B)^{c} =A^c \cap B^c, \end{eqnarray}\]
where:
- Left hand side: \((A\cup B)^{c}\) represents the set of all elements that are neither \(A\) nor \(B\);
- Right hand side: \(\color{blue}{A^c \cap B^c}\) represents the intersection of all elements that are not in \(A\) (namely they are \(A^c\)) and not \(B\) either (namely they are \(B^c\)) \(\Rightarrow\) set of all elements that are neither \(A\) nor \(B\).
Put in words, this law says that
the complement of the union between two sets is the intersection of their complements.
2.5.3 De Morganâs Theorem
We can extend these laws to three sets. Let us consider three sets \(A_{1}\), \(A_{2}\) and \(A_{3}\) in \(S\), and use the overline notation to denote the complement of a set, (i.e. \(A^{c}_{1} = \overline{A_{1}})\),
The First law then becomes: \[\overline{\left(A_{1}\cup A_{2}\cup A_{3}\right)} = \overline{A_{1}} \cap \overline{A_{2}} \cap \overline{A_{3}}\]
While the Second Law becomes: \[\overline{\left(A_{1}\cap A_{2}\cap A_{3}\right)} = \overline{A_{1}} \cup \overline{A_{2}} \cup \overline{A_{3}}\]
More generally, this can be extended to the union of an infinite amount of sets in a theorem that we present without demonstration. Consider the following notation for the union of an infinity of sets :
\[\bigcup_{i \in \mathbb{N}} A_{i} = A_{1} \cup A_{2} \cup A_{3} \cup \dots \]
and the intersection of an infinite but countable amount of sets
\[\bigcap_{i \in \mathbb{N}} A_{i} = A_{1} \cap A_{2} \cap A_{3} \cap \dots\]
2.6 Probability as Frequency
When assessing any experiment with an uncertain outcome the, primary interest is not necessarily in the events itself (as they may or may not happen) but in the degree of likelihood that it can happen. That is, we are interested in the probability of the event happening
Intuitively, the probability of an event is a value associated with the event: \[\text{event} \rightarrow \text{pr(event)}\] and it is such that:
- the probability is positive or more generally non-negative (it can be zero);
- the \(\text{pr}(S)=1\), where \(S\) is the sample space and \(\text{pr}(\varnothing)=0\);
- the probability of two (or more) mutually exclusive events is the sum of the probabilities of each event.
In many experiments, it is natural to assume that all outcomes in the sample space (\(S\)) are equally likely to occur. Think of when you throw a well balanced die, for example. If the die is well balanced, you would assume that all 6 numbers have an equal chance of happening.
Let us apply this logic in the abstract with an experiment whose sample space is a finite set, say, \(S = \{1, 2, 3, \dots N\}\). If you believe that all outcomes are equally likely to occur, you are assuming that the probability of every event, every subset with one element has the same probability:
\[P(\{1\})=P(\{2\})=...=P(\{N\})\] or equivalently \(P(\{i\})= 1/N\), for \(i=1,2,...,N\).
Now, if we define a composite event \(A\), a subset of \(S\), composed of \(N_A\) realizations i.e. elements of \(S\) having the same likelihood (namely, the same probability). Then, the probability of the event \(A\) becomes the ratio of the outcomes in \(A\) with respect to the total possible outcomes.
\[\begin{equation} \boxed{P(A)=\frac{N_A}{N}=\frac{\mbox{# of favorable outcomes}}{\mbox{total # of outcomes}}=\frac{\mbox{# of outcomes in $A$}}{\mbox{# of outcomes in $S$}}} \tag{2.1} \end{equation}\]
where the âhashâ â\(\#\)â stands for ânumberâ.
Building on the intuition gained in the last example and formula (2.1), we state a first informal definition, which establishes the probability of an event in terms of relative frequency.
Figure 2.10: Taken from Ross (2014)
Clearly, \[0 \leq n(A) \leq n, \quad \text{so} \quad 0 \leq P(A) \leq 1.\] Thus, we say that the probability is a set function (it is defined on sets) and it associates to each set/event a number between zero and one.
To express the probability, we need to impose some additional conditions, that we are going to call axioms.
We here briefly state the ideas, then we will formalize them:
When we define the probability we would want the have a domain such that it includes the sample space \(S\) and \(P(S)=1\).
Moreover, for the sake of completeness, if \(A\) is an event and we can talk about the probability that \(A\) happens, then it is suitable for us that \(A^c\) is also an event, so that we can talk about the probability that \(A\) does not happen.
Similarly, if \(A_1\) and \(A_2\) are two events (so we can say something about their probability of happening), so we should be able to say something about the probability of the event \(A_1 \cup A_2\).