1. EQT 272 PROBABILITY AND STATISTICS NORNADIA MOHD YAZID INSTITUT E OF ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS

2. CHAPTER 1: PROBABILITY 1.1 Introduction1.2 Sample space and algebra of sets1.3 Properties of probability1.4 Tree diagrams and counting techniques1.5 Conditional probability1.6 Bayes’s theorem1.7 Independence

3.  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.

4. 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

5. The mathematical basis of probability is the theory of sets. An experiment that can result in different outcomes, even though it is repeated in the same manner every time. Random experiments collection of elements or components Sets The set of all possible outsomes of random experiment. Sample space, S a subset of the sample space Events

6. Suppose that three items are selected at random from a manufacturing process. Each item is inspected and classified defective, D, or nondefective, N. Sample space : S ={ DDD, DDN, DND, DNN, NDD, NDN, NND, NNN }

8. Venn diagram Used to depicts all the possible outcomes for an experiment.

9. The union of events A and B - the set of all elements that belong to A or B or both. Meaning: joining, addition. Denoted as Union / “Or” Statement: AB

10. The intersection of events A and B - the set of all elements that belong to both A and B Meaning: overlap, things in common. Denoted by. Intersection / “And” Statement: AB

11. The complement of the event A - the event that contains all of the elements that do not belong to an event A. Meaning: not A. Denoted by. Complement:

12. When A and B have no outcomes in common, they are said to be mutually exclusive / disjoint sets. Mutually Exclusive / Disjoint:

13. EXAMPLE 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

14. A NSWER

15. 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).

17. 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 )

18. S OLUTION 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)

19. S OLUTION 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)}

20. Consider randomly selecting a UniMAP Master Degree international student. 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?

21. S OLUTION

22. 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

23. 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/Gende r Male (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?

24. SOLUTION EXAMPLE 1.4

25. 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.

26. 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.

27. E XAMPLE 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

28. S OLUTION 1. Since the two probabilities are not same, events A and D are dependent. 2. Since, events A and D are dependent.

29. M ULTIPLICATIVE R ULE OF P ROBABILITY :

30. 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

32. S OLUTION

33. - Used to revise previously calculated probabilities based on new information. - Extension of conditional probability

34. Suppose someone told you they had a nice conversation with someone on the train. Not knowing anything else about this conversation, the probability that they were speaking to a woman is 50%. Now suppose they also told you that this person had long hair. It is now more likely they were speaking to a woman, since women are more likely to have long hair than men. Bayes' theorem can be used to calculate the probability that the person is a woman.

35. Suppose it is also known that 75% of women have long hair. Likewise, suppose it is known that 15% of men have long hair. Our goal is to calculate the probability that the conversation was held with a woman, given the fact that the person had long hair.

36. Probability that the conversation was held with a woman, given the fact that the person had long hair

38. A drilling company has estimated a 40% chance of striking oil for their new well. A detailed test has been scheduled for more information. Historically, 60% of successful wells have had detailed tests, and 20% of unsuccessful wells have detailed tests. Given that this well has been scheduled for a detailed test, what is the probability that the well will be successful?

39. S OLUTION

40. TRY!!! You have a database of 100 emails. 60 of those 100 emails are spam 48 of those 60 emails that are spam have the word "buy" 12 of those 60 emails that are spam don't have the word "buy" 40 of those 100 emails aren't spam 4 of those 40 emails that aren't spam have the word "buy" 36 of those 40 emails that aren't spam don't have the word "buy" What is the probability that an email is spam if it has the word "buy"?

41. Tree diagrams help us to understand probability concepts by presenting them visually. In a tree diagram, each outcome is represented by a branch of the tree. A tree diagram helps to find simple events. 1.4.1 Tree diagrams

42. 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. Y1R1 R2

43. First ball Second ball RESULT S Y1 R1 R2 R1 R2 Y1 R2 Y1 R1 Y1R1 Y1R2 R1Y1 R1R2 R2Y1 R2R1

44. We can use counting techniques or counting rules to 1.4.2 Counting technique # find the number of ways to accomplish the experiment# find the number of simple events.# find the number of outcomes

45. Counting rules Multiplication Principle PermutationsCombinations

46. Assume an operation can be described as a sequence of k steps,  the number of ways completing step 1 is n 1  the number of ways completing step 2 is n 2  the number of ways completing step 3 is n 3 Example The design for a website is to consist of 4 colours, 3 fonts, and 3 position for an image. In order to identify the possible design, multiplication rule can be applied.

47. A permutaion of elements is an ordered sequence of the elements. All possible arrangements of a collection of things, where the order is important. There are basically two types of permutation: 1) Repetition is Allowed 2) No Repetition

48. A ) R EPETITION IS A LLOWED When you have n things to choose from... you have n choices each time! When choosing r of them, the permutations are: n × n ×... (r times) (In other words, there are n possibilities for the first choice, THEN there are n possibilites for the second choice, and so on, multplying each time.) Which is easier to write down using an exponent of r : n × n ×... (r times) = n r

49. Example: In a lock, there are 10 numbers to choose from (0,1,..9) and you choose 3 of them: 10 × 10 ×... (3 times) = 10 3 = 1,000 permutations

50. B ) N O R EPETITION In this case, you have to reduce the number of available choices each time. For example, what order could 16 pool balls be in? After choosing a ball, you can't choose it again.

51. So, your first choice would have 16 possibilites, and your next choice would then have 15 possibilities, then 14, 13, etc. And the total permutations would be: 16 × 15 × 14 × 13 ×... = 20,922,789,888,000 But maybe you don't want to choose them all, just 3 of them, so that would be only: 16 × 15 × 14 = 3,360 In other words, there are 3,360 different ways that 3 pool balls could be selected out of 16 balls.

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

53. 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.

54. "The password of the safe was 472". We do care about the order. "724" would not work, nor would "247". It has to be exactly 4-7-2. To help you to remember, think " P ermutation... P osition"

55. 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

56. ABACACBABACACBC AB AC BC BA CA CB S OLUTION

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

58. A collection of things, in which the order does not matter. Example: You are making a sandwich. How many different combinations of 2 ingredients can you make with cheese, mayo and ham? Answer: {cheese, mayo}, {cheese, ham} or {mayo, ham}

59. Formula: It is often called "n choose r"

60. 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

61. ABACAC B AB AC BC S OLUTION

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