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 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

9. Free Powerpoint Templates Page 9 Basic Operations Figure 1.1: Venn diagram representation of events

10. Free Powerpoint Templates Page 10 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

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, 5,7,9, 11,….}, the set of odd numbers Find, and

12. Free Powerpoint Templates Page 12 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

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14. Free Powerpoint Templates Page 14 Theorem 1.1 : Laws of Probability

15. Free Powerpoint Templates Page 15 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)

16. Free Powerpoint Templates Page 16 Solutions 1.2 a) Sample space, S 123456 1(1, 1)(1, 2)(1, 3)(1, 2)(1, 5)(1, 6) 2(2, 1)(2, 2)(2, 3)(2, 4)(2, 5)(2, 6) 3(3, 1)(3, 2)(3, 3)(3, 4)(3, 5)(3, 6) 4(4, 1)(4, 2)(4, 3)(4, 4)(4, 5)(4, 6) 5(5, 1)(5, 2)(5, 3)(5, 4)(5, 5)(5, 6) 6(6, 1)(6, 2)(6, 3)(6, 4)(6, 5)(6, 6)

17. Free Powerpoint Templates Page 17 Solutions 1.2 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)}

18. Free Powerpoint Templates Page 18 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?

19. Free Powerpoint Templates Page 19 Solutions 1.3

20. Free Powerpoint Templates Page 20 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

21. Free Powerpoint Templates Page 21 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?

22. SOLUTION EXAMPLE 1.4

23. Free Powerpoint Templates Page 23 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

24. Free Powerpoint Templates Page 24 Let, A = automatic transmission B = black C = grey Calculate;

25. 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

26. 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

27. Free Powerpoint Templates Page 27 Definition :  Two events in independent if and only if the probability of event B is not influenced or changed by the occurrence of event A, or vice versa  Two events A and B are said to be independent if and only if either Otherwise, the events are said to be dependent.

28. Free Powerpoint Templates Page 28 Example :- Suppose there are two children have eye brown color. Since the eye color of a child is affected by the genetic of parents and not affect by the other child, it is reasonable to assume that event A : the first child has brown eyes and B: the second child has brown eyes, are independent.

29. Example 1.5 Survey of 1000 adults, the respondents were classified according to whether they currently had a child in college and whether the loan burden for college is too high or the right amount. 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

30. Solution 1. Since the two probabilities are not same, events A and D are dependent. 2. Since, events A and D are dependent.

31. Free Powerpoint Templates Page 31 Multiplicative Rule of Probability:

32. Free Powerpoint Templates Page 32 Mutually Exclusive VS Independent Mutually ExclusiveIndependent DefinitiomTwo events cant occur together Occurrence 1 event does not effect the occurrence of another event Multiplication Rule Additional Rule Extra

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34. Free Powerpoint Templates Page 34 Solution

35. Free Powerpoint Templates Page 35 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

36. Free Powerpoint Templates Page 36 Tree Diagram When Toss A Coin

37. Free Powerpoint Templates Page 37 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

38. Free Powerpoint Templates Page 38 First ballSecond ballRESULTS W1 B1 B2 B1 B2 W1 B2 W1 B1 W1B1 W1B2 B1W1 B1B2 B2W1 B2B1

39. 1. Suppose that (a)Find (ans : 0.3) (b)Are events A and B are mutually exclusive? (ans: no) (c)If are events A and B independently? (ans: yes) 2. Let A and B be events such that : Are A and B independent? (ans: no) 3. Suppose that. (ans : 0.08, 0.52) If events A and B are independent, find probabilities : Exercise 1.6

40. 4. An electric shop sells 2 DVD player models namely L and S by cash (C) and installment (I). Last year, out of 150 units sold by cash, 30 units were from L model. At the same time, from 40 units sold by installment, 35 units were from S model. (a)Simplify the given information using a contigency table. (b)If a person bought DVD player last year selected at random, find probability that : (i) he does not buy L model (ii) he bought L model and used installment package (iii) he bought S model, given that he paid cash (c) Are the events installment and S model mutually exclusive? Prove it. Exercise 1.6

41. Free Powerpoint Templates Page 41 Exercise 1.6 5.During a space shot the primary computer is backed up by two secondary systems. They operate independently. We are interested to check the readiness of these three system at launch time (whether every system is operate (Y) or not (N)). To generate the sample space, build a tree diagram, and list all the sample space. Let events A: primary system operable B: primary system inoperable Do event A and B mutually exclusive? 6.Two dice are thrown together. Use tree diagram to find the probability that both numbers are less than 5. (ans: 4/9)

42. Free Powerpoint Templates Page 42 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

43. Free Powerpoint Templates Page 43 Counting rules MultiplicationPermutationsCombinations

44. Free Powerpoint Templates Page 44 Suppose that an experiment has n outcomes and for each of these possible outcomes the second experiment has n possible outcomes and third experiment has n possible outcomes and so forth. Then the experiment consists of performing a series of k experiment has outcomes. Also name as mn rule. (m = ways, n = second stage ways)

45. Example Suppose that order a car in one of three styles and in one of four paint colors. Find out how many options you have : mn Rule m = 3 styles n = 4 paint colors mn = (3)(4) =12 possible options (outcomes)

46. Free Powerpoint Templates Page 46 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.

47. Free Powerpoint Templates Page 47 The number of ways to arrange an entire set of n distinct items is

48. Free Powerpoint Templates Page 48 Two types of permutation: (a)Repetition is Allowed (b)No Repetition

49. Free Powerpoint Templates Page 49 Repetition is Allowed -When you have n things to choose from.. You have n choices each time -When choosing r of them, the permutations are: n x n x … (r times) (in other words, there are possibilities for the first choice, then there are n possibilities for the second choice and so on multiplying each time) - Easier to write as

50. Free Powerpoint Templates Page 50 Example: In a lock, there are 10 numbers to choose from (0,...,9) and choose 3 from them. 10 x 10 x 10 x … (3 times)

51. Free Powerpoint Templates Page 51 No repetition -In this case, you have to reduce the number of available choices each time. For example : Choose 3 balls from 16 pool balls 16 x 15 x 14 = 3360 (After choose a ball, you cant choose it again)

52. Free Powerpoint Templates Page 52 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

53. Free Powerpoint Templates Page 53 ABACACBABACACBC 1.AB 2.AC 3.BC 4.BA 5.CA 6.CB

54. Free Powerpoint Templates Page 54 There are 6 ways to select and arrange the books in order.

55. Free Powerpoint Templates Page 55 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.

56. Free Powerpoint Templates Page 56 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

57. Free Powerpoint Templates Page 57 ABACAC B 1.AB 2.AC 3.BC

58. Free Powerpoint Templates Page 58 There are 3 ways to select and arrange the books when the order is not important

59. A company produces 10 microchips during a night a staff. 6 of these turn out to be defective. Suppose 3 of chips were sent to customer. What is the probability of the customer receiving 2 defective chips?

60. Free Powerpoint Templates Page 60 Exercise 1.7 1. A candy dish contains one yellow and two red candies. Two candies are selected one at a time from the dish and their colors are recorded. How many options have? ans : 6 2.A piece of equipment is composed five parts that can be assembled in any order. A test is to be conducted to determine the time necessary for each order of assembly. If each order is to be tested once, how many tests must be conducted? ans : 120 3. In how many ways can an animal trainer arrange 5 lions and 4 tigers in a row so that no two lions are together? ans: 2880

61. Free Powerpoint Templates Page 61 4.A special type of password consists of four different letters of the alphabet, where each letter is used only once. How many different possible passwords are there? ans: 358800 5.Assuming that any arrangement of letters forms a ‘word’, how many words of any length can be formed from the letters of the word SQUARE? ans : 1956

62. Free Powerpoint Templates Page 62 Exercise 1.8 1. 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. 2.A printed circuit board may be purchased from five suppliers. In how many ways can three suppliers be chosen from the five? ans : 10 3. Ali is the Chairman of a committee. In how many ways can be a committee of 5 be chosen from 10 people given that Ali must be one of them? ans : 126

63. Free Powerpoint Templates Page 63 4.Out of 5 computer engineers, 3 communication engineers and 6 network engineers, a committee consisting of 6 engineers need to be formed. In how many ways can this be done if : (a)Any engineers can be in committee ans : 3003 (b) Out of 6 engineers, only one engineer from the field of communication can be chosen ans : 1386 5.A committee of 5 persons is to be formed from 6 men and 4 women. In how many ways can this be done when (a) at least 2 women included? ans : 186 (b) at most two women are included? ans : 186