week 2 1

Basic Set Theory

• A set is a collection of elements. Use capital letters, A, B, C to denotes sets and small letters a1, a2, … to denote the elements of the set.

• Notation: means the element a1 is an element of the set A

A = {a1, a2, a3 }.

• The null, or empty set, denoted by Ф, is the set consisting of no points. Thus, Ф is a sub set of every set.

• The set S consisting of all elements under consideration is called the universal set.

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week 2 2

Relationship Between Sets

• Any two sets A and B are equal if A and B has exactly the

same elements. Notation: A=B.

• Example: A = {2, 4, 6}, B = {n; n is even and 2 ≤ n ≤ 6}

• A is a subset of B or A is contained in B, if every point in A is

also in B. Notation:

• Example: A = {2, 4, 6}, B = {n; 2 ≤ n ≤ 6} = {2, 3, 4, 5, 6}

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week 2 3

Venn Diagram

• Sets and relationship between sets can be described by using

Venn diagram.

• Example: We toss a fair die. What is the universal set S? …

week 2 4

Union and Intersection of sets

• The union of two sets A and B, denoted by , is the set of all points that are in at least one of the sets, i.e., in A or B or both.

• Example 1: We toss a fair die…

• The intersection of two sets A and B, denoted by or AB, is the set of all points that are members of both A and B.

• Example 2: The intersection of A and B as defined in example 1 is …

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week 2 5

Properties of unions and intersections

Unions and intersections are:

• Commutative, i.e., AB = BA and

• Associative, i.e.,

• Distributive, i.e.,

• These laws also apply to arbitrary collections of sets (not just pairs).

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week 2 6

Disjoint Events

• Two sets A and B are disjoint or mutually exclusive if they

have no points in common. Then .

• Example 3: Toss a die. Let A = {1, 2, 3} and B = {4, 5}.

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week 2 7

Complement of a Set

• The complement of a set, denoted by Ac or A’ makes sense only with respect to some universal set. Ac is the set of all points of the universal set S that are not in A.

• Example: the complement of set A as defined in example 3 is…

• Note: the sets A and Ac are disjoint.

week 2 8

De Morgan’s Laws

• For any two sets A and B:

• For any collection of sets A1, A2, A3, … in any universal set S

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week 2 9

Finite, Countable Infinite and Uncountable

• A set A is finite if it contains a finite number of elements.

• A set A is countable infinite if it can be put into a one-to-one correspondence with the set of positive integers N.

• Example: the set of all integers is countable infinite because …

• The whole interval (0,1) is not countable infinite, it is uncountable.

week 2 10

The Probability Model

• An experiment is a process by which an observation is made.

For example: roll a die 6 times, toss 3 coins etc.

• The set of all possible outcomes of an experiment is called the sample space and is denoted by Ω.

• The individual elements of the sample space are denoted by ω and are often called the sample points.

• Examples ...

• An event is a subset of the sample space. Each sample point is a simple event.

• To define a probability model we also need an assessment of the likelihood of each of these events.

week 2 11

σ – Algebra

• A σ-algebra, F, is a collection of subsets of Ω satisfying the following properties:F contains Ф and Ω.F is closed under taking complement, i.e., F is closed under taking countable union, i.e.,

• Claim: these properties imply that F is closed under countable intersection.

• Proof: …

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week 2 12

Probability Measure A probability measure P mapping F [0,1] must satisfy

• For , P(A) ≥ 0 .

• P(Ω) = 1.

• For , where Ai are disjoint,

This property is called countable additivity.

• These properties are also called axioms of probability.

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week 2 13

Formal Definition of Probability Model

• A probability space consists of three elements (Ω, F, P)

(1) a set Ω – the sample space.

(2) a σ-algebra F - collection of subsets of Ω.

(3) a probability measure P mapping F [0,1].

week 2 14

Discrete Sample Space

• A discrete sample space is one that contains either a finite or a countable number of distinct sample points.

• For a discrete sample space it suffices to assign probabilities to each sample point.

• There are experiments for which the sample space is not countable and hence is not discrete. For example, the experiment consists of measuring the blood pressure of patients with heart disease.

week 2 15

Calculating Probabilities when Ω is Finite

• Suppose Ω has n distinct outcomes, Ω = {ω1, ω2,…, ωn}. The probability of an event A is

• In many situations, the outcomes of Ω are equally likely, then,

• Example, when rolling a die for i = 1, 2, …, 6.

• In these situations the probability that an event A occurs is

• Example:

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week 2 16

Rules of Probability

• for all

• Corollary:

• The probability of the union of any two events A and B is

Proof: …

• If then,

Proof:

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week 2 17

• Inclusion / Exclusion formula:

For any finite collection of events

• For any finite collection of events

Proof: By induction

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