EQT 272 PROBABILITY AND STATISTICS ROHANA BINTI ABDUL HAMID INSTITUT E FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS Free Powerpoint Templates

CHAPTER 1 PROBABILITY 1.1 Introduction 1.2 Sample space and algebra of sets 1.3 Tree diagrams and counting techniques 1.4 Properties of probability 1.5 Conditional probability 1.6 Bayes’s theorem 1.7 Independence

1.1 INTRODUCTION TO PROBABILITY WHY DO COMPUTER ENGINEERS NEED TO STUDY PROBABILITY??????? Signal processing Computer memories Optical communication systems Wireless communication systems Computer network traffic

1.1 INTRODUCTION TO PROBABILITY Probability and statistics are related in an important way. Probability is used as a tool; it allows you to evaluate the reliability of your conclusions about the population when you have only sample information.

1.1 INTRODUCTION TO PROBABILITY Probability is a measure of the likelihood of an event A occurring in one experiment or trial and it is denoted by P (A).

1.1 INTRODUCTION TO PROBABILITY Experiment An experiment is any process of making an observation leading to outcomes for a sample space. Example: -Toss a die and observe the number that appears on the upper face. A medical technician records a person’s blood type. Recording a test grade.

2. Sample space and algebra of sets The mathematical basis of probability is the theory of sets. Sets A set is a collection of elements or components Sample Spaces, S A sample space consists of points that correspond to all possible outcomes. Events An event is a set of outcomes of an experiment and a subset of the sample space.

Example 1.1 Experiment: Tossing a die Sample space: Events: A: Observe an odd number B: Observe a number less than 4 C: Observe a number which could divide by 3

1.2 Sample space and algebra of sets Basic Operations   Figure 1.1: Venn diagram representation of events S B A C

1.2 Sample space and algebra of sets The union of events A and B, which is denoted as , is the set of all elements that belong to A or B or both. Two or more events are called collective exhaustive events if the unions of these events result in the sample space.   2. The intersection of events A and B, which is denoted by , is the set of all elements that belong to both A and B. When A and B have no outcomes in common, they are said to be mutually exclusive or disjoint sets. 3. The event that contains all of the elements that do not belong to an event A is called the complement of A and is denoted by

Exercise 1.1 Given the following sets; A= {2, 4, 6, 8, 10} B= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} C= {1, 3, 5, 11,….}, the set of odd numbers Find , and

Answer = {1, 2, 3, 4, 5, 6, 7, 8, 9,10} = {2, 4, 6, 8, 10} = {2, 4, 6, 8,…}, the set of even numbers

1.3 tree diagrams and counting technique Some experiments can be generated in stages, and the sample space can be displayed in a tree diagram. Each successive level of branching on the tree corresponds to a step required to generate the final outcome. A tree diagram helps to find simple events.

Example 1.2 A box contains one yellow and two red balls. Two balls are randomly selected and their colors recorded. Construct a tree diagram for this experiment and state the simple events. Y1 R1 R2

First ball Second ball RESULTS Y1R1 Y1R2 R1Y1 R1R2 R2Y1 R2R1 Y1 R1 R2

Exercise 1.2 3 people are randomly selected from voter registration and driving records to report for jury duty. The gender of each person is noted by the county clerk. List the simple events by creating a tree diagram.

1.3 tree diagrams and counting technique We can use counting techniques or counting rules to # find the number of ways to accomplish the experiment # find the number of simple events. # find the number of outcomes

Counting rules Permutations Combinations

Permutations This counting rule count the number of outcomes when the experiment involves selecting r objects from a set of n objects when the order of selection is important.

Permutations The number of ways to arrange an entire set of n distinct items is

Example 1.3 Suppose you have 3 books, A, B and C but you have room for only two on your bookshelf. In how many ways can you select and arrange the two books when the order is important. A B C

A B A B 1.AB 4.BA A C A C 5.CA 2.AC A C B A C B 3.BC 6.CB

There are 6 ways to select and arrange the books in order.

Exercise 1.3 Three lottery tickets are drawn from a total of 50. If the tickets will be distributed to each of the employees in the order in which they are drawn, the order will be important. How many simple events are associated with the experiment?

Combinations This counting rule count the number of outcomes when the experiment involves selecting r objects from a set of n objects when the order of selection is not important.

Example 1.4 Suppose you have 3 books, A, B and C but you have room for only two on your bookshelf. In how many ways can you select and arrange the two books when the order is not important. A B C

A B 1.AB A C 2.AC A C B 3.BC

There are 3 ways to select and arrange the books when the order is not important

Exercise 1.4 Suppose that in the taste test, each participant samples 8 products and is asked the 3 best products, but not in any particular order. Calculate the number of possible answer test.

1.4 PROPERTIES OF PROBABILITY

Theorem 1.1 : Laws of Probability

Example 1.5 Two fair dice are thrown. Determine   a) the sample space of the experiment b) the elements of event A if the outcomes of both dice thrown are showing the same digit. c) the elements of event B if the first thrown giving a greater digit than the second thrown. d) probability of event A, P(A) and event B, P(B)

Solutions 1.5 a) Sample space, S 1 2 3 4 5 6 (1, 1) (1, 2) (1, 3) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)

Solutions 1.5 b) A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)} c) B = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3), (5, 4), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}

Example 1.6 Consider randomly selecting a UniMAP Master Degree international student, and let A denote the event that the selected individual has a Visa Card and B has a Master Card. Suppose that P(A) = 0.5 and P(B) = 0.4 and = 0.25. a) Compute the probability that the selected individual has at least one of the two types of cards ? b) What is the probability that the selected individual has neither type of card?  

Solutions 1.6

1.5 CONDITIONAL PROBABILITY Definition: For any two events A and B with P(B) > 0, the conditional probability of A given that B has occurred is defined by

Example 1.7 A study of 100 students who get A in Mathematics in SPM examination was done by UniMAP first year students. The results are given in the table : Area/Gender Male (C) Female (D) Total Urban (A) 35 10 45 Rural (B) 25 30 55 60 40 100 If a student is selected at random and have been told that the individual is a male student, what is the probability of he is from urban area?

Example 1.8 In 2006, Edaran Automobil Negara (EON) will produce a multipurpose national car (MPV) equipped with either manual or automatic transmission and the car is available in one of four metallic colours. Relevant probabilities for various combinations of transmission type and colour are given in the accompanying table: Transmission type/Colour Grey (C) Blue Black (B) Red Automatic, (A) 0.15 0.10 Manual 0.05 0.20

Example 1.8 Let, A = automatic transmission B = black C = grey   Calculate;

1.6 BAYES’ THEOREM

Example 1.9 There are three boxes: Box 1 contains one red ball and three white balls; box 2 contains two red balls and two white balls; box 3 contains three red balls and one white ball. A box is selected at random and then a ball is chosen at random from the selected box. Determine the conditional probability that box 1 was selected, given that red ball is chosen.

1.7 independence Definition : Two events A and B are said to be independent if and only if either Otherwise, the events are said to be dependent.

Multiplicative Rule of Probability:

Example 1.10