Statistical Experiments

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 = {e1, e2, e3, …, en} • where ei = each outcome of interest EGR 252.001 2011

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} EGR 252.001 2011

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… } EGR 252.001 2011

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} EGR 252.001 2011

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} EGR 252.001 2011

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 EGR 252.001 2011

7. Other Venn Diagram Examples • Five non-mutually exclusive events • Subset – The green circle is a subset of the beige circle EGR 252.001 2011

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 EGR 252.001 2011

9. Sample Points • Multiplication Rule • If event A can occur n1ways and event B can occur n2ways, then an event C that includes both A and B can occur n1 n2ways. • 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 . EGR 252.001 2011

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 EGR 252.001 2011

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 EGR 252.001 2011

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 12P3 = 12!/(12-3)! = 1320 EGR 252.001 2011

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 12C3 = 12! / [3!(12-3)!] = 220 EGR 252.001 2011

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(A1U A2U … U Ak) = P(A1) + P(A2) + … P(Ak) EGR 252.001 2011

15. Calculating Probabilities • Examples: • 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 EGR 252.001 2011

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)? EGR 252.001 2011

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 EGR 252.001 2011

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) = EGR 252.001 2011