INTRODUCTION TO FUNDAMENTALS OF PROBABILITY

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We here present the basic concepts of probability theory using the set-theoretic approach. Thus we shall first define the basic concepts of sets.

Basic Concepts of Sets

A set is a collection of objects. Sets are usually designated by capital letters, A, B, C, etc.

The members of the set, Say A, are called the elements of the set. In describing which objects are contained in a set, three methods are available.

  1. We may list the members of the set. For example,
    A = { 1, 2, 3, 3}
    describes the set of positive integers 1, 2, 3, and 4.
  2. We may describe the set in words. For example. set contains all integers number 1 and 4, includes and
  3. We may simply write
    A = { x | x ∈ Ζ ; j ≤ x ≤ 4 }.

When “x” is a member of set A we write x ∈ A and when “x” not a member of A we write x ∉ A.

The universal set is a set of all objects under construction and is generally denoted by U. A set having no element is called a null set or an empty set and is usually denoted by Φ.

If two sets are considered, say A and B, we say B is a subset of A, If every element in A is also an element of B, and we denoted A ⊂ B.

The set A and B are said to be equal(i.e A = B ) if and only A ⊂ B and B ⊂ A.

From the above description of sets, the following consequences are immediate:

  1. For any set A, the empty or null set is a subset of A; i.e ∅ ⊂ A
  2. Once the universal set U, we have A ⊂ U
  3. For a given set A, A ⊂ A (a reflexive relation). This implies that every set is a subset of itself.
  4. If A ⊂ B and B ⊂ D, then A ⊂ D (a transitive relation).

Operations of Sets

Let A and B be any subsets of the universal set U. Then

  1. The complement of A (with respect to U) is the set made up of the elements of U, that does not belong to A. It is usually denoted as Ȧ̇′ or A̅ or Ac.
  2. The intersection of A and B is the set of elements that belongs to both set A and B is the set of elements that belong to both set A and B and it is usually denoted by A ∩ B.
  3. The union of A and B is a set of elements that belong to at least one of the sets A or B or both. It is usually denoted by  A ∩ B.

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