Statistical Experiments The set of all possible outcomes

Statistical Experiments § The set of all possible outcomes of an experiment is the Sample Space, S. § Each outcome of the experiment is an element or member or sample point. § If the set of outcomes is finite, the outcomes in the sample space can be listed as shown: § S = {H, T} § S = {1, 2, 3, 4, 5, 6} § in general, S = {e 1, e 2, e 3, …, en} § where ei = each outcome of interest JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 1

Tree Diagram § If the set of outcomes is finite sometimes a tree diagram is helpful in determining the elements in the sample space. § The tree diagram for students enrolled in the School of Engineering by gender and degree: S M EGR IDM F TCO EGR IDM TCO § The sample space: S = {MEGR, MIDM, MTCO, FEGR, FIDM, FTCO} JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 2

Your Turn: Sample Space § Your turn: The sample space of gender and specialization of all BSE students in the School of Engineering is … S = {FECE, MECE, FEVE, MEVE, FISE, MISE, FMAE, etc} or S = {BMEF, BMEM, CPEF, CPEM, ECEF, ECEM, ISEF, ISEM… } § 2 genders, 6 specializations, § 12 outcomes in the entire sample space JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 3

Definition of an Event § A subset of the sample space reflecting the specific occurrences of interest. § Example: In the sample space of gender and specialization of all BSE students in the School of Engineering, the event F could be “the student is female” § F = {BMEF, CPEF, ECEF, EVEF, ISEF, MAEF} JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 4

Operations on Events § Complement of an event, (A’, if A is the event) § If event F is students who are female, F’ = {BMEM, EVEM, CPEM, ECEM, ISEM, MAEM} § Intersection of two events, (A ∩ B) § If E = environmental engineering students and F = female students, (E ∩ F) = {EVEF} § Union of two events, (A U B) § If E =environmental engineering students and I = industrial engineering students, (E U I) = {EVEF, EVEM, ISEF, ISEM} JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 5

Venn Diagrams § Mutually exclusive or disjoint events Male Female § Intersection of two events Let Event E be EVE students (green circle) Let Event F be female students (red circle) E ∩ F is the overlap – brown area JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 6

Other Venn Diagram Examples § Five non-mutually exclusive events § Subset – The green circle is a subset of the beige circle JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 7

Subset Examples § § Students who are male Students who are ECE Students who are on the ME track in ECE Female students who are required to take ISE 428 to graduate § Female students in this room who are wearing jeans § Printers in the engineering building that are available for student use JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 8

Sample Points § Multiplication Rule § If event A can occur n 1 ways and event B can occur n 2 ways, then an event C that includes both A and B can occur n 1 n 2 ways. § Example, if there are 6 different female students and 6 different male students in the room, then there are 6 * 6 = 36 ways to choose a team consisting of a female and a male student. JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 9

Permutations § Definition: an arrangement of all or part of a set of objects. § The total number of permutations of the 6 engineering specializations in MUSE is … 6*5*4*3*2*1 = 720 § In general, the number of permutations of n objects is n! NOTE: 1! = 1 and 0! = 1 § How would the number change if we considered 7 specializations? JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 10

Permutation Subsets § In general, where n = the total number of distinct items and r = the number of items in the subset § Given that there are 6 specializations, if we take the number of specializations 3 at a time (n = 6, r = 3), the number of permutations is JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 11

Permutation Example § A new group, the MUSE Ambassadors, is being formed and will consist of two students (1 male and 1 female) from each of the 6 BSE specializations. If a prospective student comes to campus, he or she will be assigned one Ambassador at random as a guide. If three prospective students are coming to campus on one day, how many possible selections of Ambassador are there? § If the outcome is defined as ‘ambassador assigned to student 1, ambassador assigned to student 2, ambassador assigned to student 3’ § Outcomes are : A 1, A 2, A 3 or A 2, A 4, A 12 or A 2, A 1, A 3 etc § Total number of outcomes is 12 P 3 = 12!/(12 -3)! = 1320 JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 12

Combinations § Selections of subsets without regard to order. § Example: How many ways can we select 3 guides from the 12 Ambassadors? § Outcomes are : A 1, A 2, A 3 or A 2, A 4, A 12 or A 12, A 1, A 3 but not A 2, A 1, A 3 § Total number of outcomes is 12 C 3 = 12! / [3!(12 -3)!] = 220 JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 13

Introduction to Probability § The probability of an event, A is the likelihood of that event given the entire sample space of possible events. § P(A) = target outcome / all possible outcomes § 0 ≤ P(A) ≤ 1 P(ø) = 0 P(S) = 1 § For mutually exclusive events, P(A 1 U A 2 U … U Ak) = P(A 1) + P(A 2) + … P(Ak) JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 14

Calculating Probabilities § Examples: 1. There are 26 students enrolled in a section of EGR 252, 3 of whom are BME students. The probability of selecting a BME student at random off of the class roll is: P(BME) = 3/26 = 0. 1154 2. The probability of drawing 1 heart from a standard 52 card deck is: P(heart) = 13/52 = 1/4 JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 15

Additive Rules Experiment: Draw one card at random from a standard 52 card deck. What is the probability that the card is a heart or a diamond? Note that hearts and diamonds are mutually exclusive. Your turn: What is the probability that the card drawn at random is a heart or a face card (J, Q, K)? JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 16

Your Turn: Solution Experiment: Draw one card at random from a standard 52 card deck. What is the probability that the card drawn at random is a heart or a face card (J, Q, K)? Note that hearts and face cards are not mutually exclusive. P(H U F) = P(H) + P(F) – P(H∩F) = 13/52 + 12/52 – 3/52 = 22/52 JMB Chapter 2 Lecture 1 v 3 EGR 252 JMB 2019 Slide 17

Card-Playing Probability Example § P(A) = target outcome / all possible outcomes § Suppose the experiment is “being dealt 5 cards from a 52 card deck” § Suppose Event A is “ 3 kings and 2 jacks” K J K K J J (combination or perm. ? ) combinations(3 kings) = § P(A) = JMB Chapter 2 Lecture 1 v 3 combinations(2 jacks) = = 9. 23 E-06 EGR 252 JMB 2019 Slide 18