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The World of Statistics and Probability ⁽¹⁾

By Dr. Magdi Abadir, PhD

Article 15: An Introduction to Probabilities - 1

1. The concept of probability

Working on wrapping sold items in a supermarket can be a real bore. That is why, Dina has decided, for a change, not only to tally the number of customers on her eight hours shift, but also to spot how many of them were wearing glasses. She counted 356 customers of whom 122 wore glasses. On her way home in the underground, she decided to calculate the percentage of glass wearing persons. To do this, she simply divided 122 by 356 and multiplied the obtained number by 100 to get about 24.3%.

This simple example defines the term “probability” in its simplest form, although it is conventionally expressed as a fraction rather than a percentage. That is, the probability of finding a glass wearing person will be 0.243. This is obtained by dividing the number of items satisfying a certain condition by the total number of items. This concept needs however to be developed in a more quantitative way. That is why, the following sections will deal with the details of probabilities calculations.

2. Sets

2.1 Basic definitions
A set is simply a collection of objects, things, numbers, or symbols which are clearly defined. These are called the elements of the set which is usually expressed as a capital letter. The following are examples of finite sets; that is sets with a limited number of elements.

C = Set of playing cards symbols = {♣, ♦, ♥, ♠}

A = Set of integers between 2 and 9 = {3,4,5,6,7,8}

F = Set of names of five family members = {Joe, May, Suzie, Bob, Ron}

It is important to notice that the order in which the elements are written in a set is immaterial and these can be placed in any order without modifying the nature of the set.

2.2 Basic terminologies

Some sets are composed of a limited number of elements. These are called finite sets, like the three examples above. On the other hand, some sets contain an infinite number of elements, like the set of all positive integers, usually denoted by Z⁺:
Z⁺ = {1,2,3,4,5,…}

The number of elements of a finite set E is called the Cardinal of the set, denoted by n(E). For example, in section 2.1, n(C) =4

An empty set is a set containing no elements; like, for example the set of months with 32 days. It is usually denoted by the symbol Q, so that n(Q) =0

A singleton is a set containing one single element like X= {3} so that n(X) =1

Consider the set P of positive integers from 1 to 6: P ={1,2,3,4,5,6}. Any element of the set like 2, for example, is said to belong to that set. This is denoted by 2 ∈ P, whereas 7 which doesn’t belong to P will be denoted by 7 ∉ P.

A set like B = {1,2,6} whose elements are contained in P is called a subset of P, which is denoted by B ⊂ P. In general, the empty set Q is a subset of any other set A: Q ⊂ A

References:
(1) H. B. Enderton, “Elements of set theory” Academic Press., Chapter 1 (1977)

Dr. Magdi Fouad Abadir, Ph. D.: Dr. M. F. Abadir is currently a professor with the Chemical Engineering Department at the Faculty of Engineering, University of Cairo, Egypt. His major interests are in the fields of high temperature science and technology. During his career, he has supervised more than 110 MSc and PhD theses and published more than a hundred papers mostly in international peer review journals. He currently teaches courses in High Temperature Technology and Industrial Statistics. He is also a consultant for several industrial businesses.

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TechTool

The World of Statistics and Probability ⁽¹⁾

By Dr. Magdi Abadir, PhD

Article 15: An Introduction to Probabilities - 1

1. The concept of probability

2. Sets

2.2 Basic terminologies

Site Map

"optimanage.com" TERMS & CONDITIONS

TechTool

The World of Statistics and Probability (1)

By Dr. Magdi Abadir, PhD

Article 15: An Introduction to Probabilities - 1

1. The concept of probability

2. Sets

2.2 Basic terminologies

Site Map

"optimanage.com" TERMS & CONDITIONS

The World of Statistics and Probability ⁽¹⁾