1. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 1 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, …, e n }  where e i = each outcome of interest

2. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 2 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:  The sample space: S = {MEGR, MIDM, MTCO, FEGR, FIDM, FTCO} S M EGRIDMTCO F EGRIDMTCO

3. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 3 Your Turn: Sample Space  Your turn: The sample space of gender and specialization of all BSE students in the School of Engineering is … or  2 genders, 6 specializations,  12 outcomes in the entire sample space S = {FECE, MECE, FEVE, MEVE, FISE, MISE, FMAE, etc} S = {BMEF, BMEM, CPEF, CPEM, ECEF, ECEM, ISEF, ISEM… }

4. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 4 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}

5. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 5 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}

6. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 6 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

7. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 7 Other Venn Diagram Examples  Five non-mutually exclusive events  Subset – The green circle is a subset of the beige circle

8. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 8 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

9. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 9 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.

10. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 10 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

11. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 11 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

12. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 12 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 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 : A1,A2,A3 or A2,A4,A12 or A2, A1,A3 etc  Total number of outcomes is 12 P 3 = 12!/(12-3)! = 1320

13. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 13 Combinations  Selections of subsets without regard to order.  Example: How many ways can we select 3 guides from the 12 Ambassadors?  Outcomes are : A1,A2,A3 or A2,A4,A12 or A12, A1,A3 but not A2,A1,A3  Total number of outcomes is 12 C 3 = 12! / [3!(12-3)!] = 220

14. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 14 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 A k ) = P(A 1 ) + P(A 2 ) + … P(A k )

15. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 15 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

16. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 16 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)?

17. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 17 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

18. JMB Chapter 2 Lecture 1 v3EGR 252 Spring 2014Slide 18 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 J K K K K J J (combination or perm.?)  P(A) = = 9.23E-06 combinations(3 kings) =combinations(2 jacks) =