1. Free Powerpoint Templates Page 1 Free Powerpoint Templates EQT 272 PROBABILITY AND STATISTICS SYAFAWATI AB. SAAD INSTITUTE FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS

2. Free Powerpoint Templates Page 2 CHAPTER 1 PROBABILITY 1.1 Introduction1.2 Sample space and algebra of sets1.3 Tree diagrams and counting techniques1.4 Properties of probability1.5 Conditional probability1.6 Independence

3. Free Powerpoint Templates Page 3 WHY DO COMPUTER ENGINEERS NEED TO STUDY PROBABILITY??????? 1.Signal processing 2.Computer memories 3.Optical communication systems 4.Wireless communication systems 5.Computer network traffic

4. Free Powerpoint Templates Page 4  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.

5. Free Powerpoint Templates Page 5 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).

6. Free Powerpoint Templates Page 6 Experiment An experiment is any process of making an observation leading to outcomes for a sample space. Example: -Toss dice and observe the number that appears on the upper face. -A medical technician records a person’s blood type. -Recording a test grade.

7. Free Powerpoint Templates Page 7 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.

8. Free Powerpoint Templates Page 8 Basic Operations Figure 1.1: Venn diagram representation of events

9. Free Powerpoint Templates Page 9 1.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

10. Free Powerpoint Templates Page 10 Experiment: Tossing a dice Sample space: S ={1, 2, 3, 4, 5, 6} Events: A: Observe an odd number B: Observe a number less than 4 C: Observe a number which could divide by 3

11. Free Powerpoint Templates Page 11 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

12. Free Powerpoint Templates Page 12

13. Free Powerpoint Templates Page 13 Theorem 1.1 : Laws of Probability

14. Free Powerpoint Templates Page 14 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)

15. Free Powerpoint Templates Page 15 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?

16. Free Powerpoint Templates Page 16 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

17. Free Powerpoint Templates Page 17 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/GenderMale (C)Female (D)Total Urban (A)351045 Rural (B)253055 Total6040100 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?

18. Free Powerpoint Templates Page 18 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)0.150.10 Manual0.150.050.150.20

19. Free Powerpoint Templates Page 19 Let, A = automatic transmission B = black C = grey Calculate;

20. EXERCISE 1.3 From a survey of 100 college students, a marketing research company found that 75 students owned stereos cars, and 35 owned cars and stereos. (a)Draw a Venn diagram. (b)Find the probability that students owned either a car or a stereo. (c)Find the probability that students did not owned either a car or a stereo.

21. Exercise 1.4 Suppose that, there are 51% men and 49% women, and that the proportions of colorblind men and women are shown in table below: (a)Find the probability wear spectacles, given that men. (b)Find the probability of wear spectacles, given that women. (c)Find the probability of not wear spectacles, given that women. Men (M)Women (W)Total Wear Spectacles (S) 0.040.0020.042 Not Wear Spectacles (NS) 0.470.4880.958 Total0.510.491.00

22. Exercise 1.5 Assuming the type distribution to be A:41%, B:9%, AB:4%, O:46%, (a)what is the probability that the blood of a randomly selected individual will contain A antigen? (b)Contain B antigen? (c)Contain neither A nor B antigen

23. Free Powerpoint Templates Page 23 Definition :  Two events A and B are said to be independent if and only if either Otherwise, the events are said to be dependent.

24. Example 1.5 Refer to table below: Are events A and D independent? Too High (A) Right Amount (B) Too Little (C) Child in College (D) 0.350.080.01 No Child in College (E) 0.250.200.11

25. Free Powerpoint Templates Page 25 Multiplicative Rule of Probability:

26. Free Powerpoint Templates Page 26

27. Free Powerpoint Templates Page 27 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 sample space. 1.3.1 Tree diagrams

28. Free Powerpoint Templates Page 28 A box contains one white and two blue balls. Two balls are randomly selected and their colors recorded. Construct a tree diagram for this experiment and state the simple events. W1B1 B2

29. Free Powerpoint Templates Page 29 Exercise 1.6 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.

30. Free Powerpoint Templates Page 30 We can use counting techniques or counting rules to 1.3.2 Counting technique # find the number of ways to accomplish the experiment # find the number of sample space.# find the number of outcomes

31. Free Powerpoint Templates Page 31 Counting rules PermutationsCombinations

32. Free Powerpoint Templates Page 32 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.

33. Free Powerpoint Templates Page 33 The number of ways to arrange an entire set of n distinct items is

34. Free Powerpoint Templates Page 34 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 A B B C C

35. Free Powerpoint Templates Page 35 Exercise 1.7 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?

36. Free Powerpoint Templates Page 36 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.

37. Free Powerpoint Templates Page 37 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 A B B C C

38. Free Powerpoint Templates Page 38 Exercise 1.8 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.