#### Transcript chapter 1 - UniMAP Portal

EQT 272 PROBABILITY AND STATISTICS ROHANA BINTI ABDUL HAMID INSTITUT E FOR ENGINEERING MATHEMATICS (IMK) UNIVERSITI MALAYSIA PERLIS Free Powerpoint Templates Page 1 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 Free Powerpoint Templates 1.7 Independence Page 2 WHY DO COMPUTER ENGINEERS NEED TO STUDY PROBABILITY??????? 1. 2. 3. 4. 5. Signal processing Computer memories Optical communication systems Wireless communication systems Computer network traffic Free Powerpoint Templates Page 3 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. Free Powerpoint Templates Page 4 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). number of ways thattheevent A can occur ( A) P( A) totalnumber of outcomes( S ) n( A) n( S ) Free Powerpoint Templates Page 5 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. Free Powerpoint Templates Page 6 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. Free Powerpoint Templates Page 7 • Experiment: Tossing a die • 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 Free Powerpoint Templates Page 8 Basic Operations S B A Figure 1.1: Venn diagram representation of events Free Powerpoint Templates Page 9 1. The union of events A and B, which is denoted as A B , - 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 A B, - 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 Freecomplement Powerpoint Templates an event A is called the of A and is denoted byPage A 10 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 A B , A B and C Free Powerpoint Templates Page 11 Answer • A B = {1, 2, 3, 4, 5, 6, 7, 8, 9,10} • A B = {2, 4, 6, 8, 10} • C = {2, 4, 6, 8,…}, the set of even numbers Free Powerpoint Templates Page 12 1.3.1 Tree diagrams • 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 Free Powerpoint Templates events. Page 13 • 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. W1 B1 B2 Free Powerpoint Templates Page 14 First ball Second ball B1 W1 B2 W1 B1 B2 RESULTS W1B1 W1B2 B1W1 B1B2 W1 B2W1 B1 B2B1 B2 Free Powerpoint Templates Page 15 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. Free Powerpoint Templates Page 16 1.3.2 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 Free Powerpoint Templates Page 17 Permutations Counting rules Combinations Free Powerpoint Templates Page 18 • 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. n n! Pr ( n r )! Free Powerpoint Templates Page 19 • The number of ways to arrange an entire set of n distinct items is n Pn n! Free Powerpoint Templates Page 20 • 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 Free Powerpoint Templates Page 21 A B A C A B C B A 2.AC C A 3.BC C A C B 1.AB Free Powerpoint Templates 4.BA 5.CA 6.CB Page 22 n 3 n! Pr ( n r )! 3! P2 ( 3 2 )! 6 There are 6 ways to select and arrange the books in order. Free Powerpoint Templates Page 23 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? Free Powerpoint Templates Page 24 • 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. n! nC n r r r ! n r ! Free Powerpoint Templates Page 25 • 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 Free Powerpoint Templates Page 26 A B 1.AB A C 2.AC A B C 3.BC Free Powerpoint Templates Page 27 n! nC n r r r ! n r ! 3 3! C2 2!( 3 2 )! 3 There are 3 ways to select and arrange the books when the order is not Free Powerpoint Templates important Page 28 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. Free Powerpoint Templates Page 29 1) 2) 3) 4) 5) 6) 7) 8) 9) 0 P ( A) 1 P ( A) P ( A) 1 P ( A B) P ( A) P ( A B ) P ( A B ) P ( B ) P ( A B ) P ( A B) 1 P ( A B ) P (( A B )) P ( A B) P (( A B )) P ( A B) P ( A ( A B )) P ( A B ) P ( B ) P[( A B ) ( A B )] S B A A B A B Free Powerpoint Templates A B Page 30 Theorem 1.1 : Laws of Probability a) P( A) 1 – P A b) P( A B) P A P B – P( A B) c) P( A B C ) P A P B P C – P( A B) – P( A C ) – P( B C ) P( A B C ) d) If A and B are mutually exclusive events, then P( A B) 0 e) If A1 and A2 are the subset of S where A1 A2 , then P A1 P A2 Free Powerpoint Templates Page 31 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) Free Powerpoint Templates Page 32 Solutions 1.5 a) Sample space, S 1 2 3 4 5 6 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) Free Powerpoint Templates Page 33 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)} n( A) 6 1 d) P A n( S ) 36 6 n( B) 15 5 P B n( S ) 36 12 Free Powerpoint Templates Page 34 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 P( A B) = 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? Free Powerpoint Templates Page 35 Solutions 1.6 a) P( A B) P A P B – P( A B ) = 0.5 0.4 – 0.25 0.65 b) 1 P( A B) 1 – 0.65 0.35 Free Powerpoint Templates Page 36 • 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 P( A B) P( A | B) P( B) Free Powerpoint Templates Page 37 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 Total 60 40 100 If a student is selected at random and have been told that the individual is a male student, what is the Free Powerpoint Templates probability of he is from urban area? Page 38 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 Black Grey (C) Blue Automatic, (A) 0.15 0.10 0.10 0.10 Manual 0.15 0.05 0.15 0.20 type/Colour Free Powerpoint Templates (B) Red Page 39 • Let, A = automatic transmission B = black C = grey Calculate; a) P ( A), P ( B ) and P( A B) b) P ( A | B ) and P( B | A) c) P ( A | C ) and P ( A | C ) Free Powerpoint Templates Page 40 If A1 , A2 ,..., An is a partition of a sample space, then the posterior probabilities of events Ai conditional on an event B can be obtained from the probabilities P Ai and P B | Ai using the formula, P Ai B P Ai P B | Ai P Ai | B P B P B P Ai P B | Ai P A PB | A n j 1 j Free Powerpoint Templates j Page 41 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. Free Powerpoint Templates Page 42 • Definition : Two events A and B are said to be independent if and only if either P ( A | B ) P ( A) or P ( B | A) P ( B ) Otherwise, the events are said to be dependent. Free Powerpoint Templates Page 43 Multiplicative Rule of Probability: The probability that both two events A and B, occur is P( A B) P A P B | A P B P A | B If A and B are independent, P( A B) P A P B Free Powerpoint Templates Page 44 3 1 Suppose that P( A) and P ( B ) . Are events A and B independent or 5 3 mutually exclusive if , 1 a) P( A B) 5 14 b) P( A B) 15 Free Powerpoint Templates Page 45