1. EGR 252 - 21 Sample Space, S The set of all possible outcomes of an experiment. Each outcome is an element or member or sample point. If the set is finite (e.g., H/T on coin toss, number on the die, etc.): –S = {H, T} –S = {1, 2, 3, 4, 5, 6} –in general, S = {e 1, e 2, e 3, …, e n } where e i = the outcomes of interest Note: sometimes a tree diagram is helpful in determining the sample space…

2. EGR 252 - 22 Sample Space Example: The sample space of gender and specialization of all BSE students in the School of Engineering is …

3. EGR 252 - 23 Events A subset of the sample space reflecting the specific occurrences of interest. Example, –All female students, F =

4. EGR 252 - 24 Events Complement of an event, (A’, if A is the event) –e.g., students who are not female, Intersection of two events, (A ∩ B) –e.g., engineering students who are EVE and female, Mutually exclusive or disjoint events Union of two events, (A U B)

5. EGR 252 - 25 Venn Diagrams Example, events V (EVE students) and F (female students)

6. EGR 252 - 26 Other Venn Diagram Examples Mutually exclusive events Subsets

7. EGR 252 - 27 Example: Students who are male, students who are ECE, students who are on the ME track in ECE, and female students who are required to take ISE 412 to graduate.

8. EGR 252 - 28 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 ways to choose a female engineering student at random and there are 6 ways to choose a male student at random, then there are 6 * 6 = 36 ways to choose a female and a male engineering student at random.

9. EGR 252 - 29 Another Example Example 2.14, pg. 32

10. EGR 252 - 210 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 … In general, the number of permutations of n objects is n!

11. EGR 252 - 211 Permutations If we take the number of specializations 3 at a time (n = 6, r = 3), the number of permutations is In general,

12. EGR 252 - 212 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?

13. EGR 252 - 213 Combinations Selections of subsets without regard to order. Example: How many ways can we select 3 guides from the 12 Ambassadors?

14. EGR 252 - 214 Probability The probability of an event, A is the likelihood of that event given the entire sample space of possible events. 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. EGR 252 - 215 Calculating Probabilities Examples: 1.There are 26 students enrolled in this 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 = ______________________ 2.The probability of being dealt 2 aces & 3 jacks in a 5- card poker hand is:

16. EGR 252 - 216 Additive Rule Examples 1.Draw 1 Card. Note Kind, Color & Suit. –Probabilities associated with drawing an ace and with drawing a black card are shown in the following contingency table: –Therefore the probability of drawing an ace or a black card is given by: Type Color Total RedBlack Ace224 Non-Ace24 48 Total26 52

17. EGR 252 - 217 Additive Rules 2.After the first card is drawn, it is returned to the deck which is shuffled. Another card is drawn. What is the probability that at least one of the cards is an ace?

18. EGR 252 - 218 Applications of Probability Example: An appliance manufacturer has learned of an increased incidence of short circuits and fires in a line of ranges sold over a 5 month period. A review of the FMEA data indicates the probabilities that if a short circuit occurs, it will be at any one of several locations is as follows: LocationP House Junction0.46 Oven/MW junction0.14 Thermostat0.09 Oven coil0.24 Electronic controls0.07

19. EGR 252 - 219 Applications of Probability The probability that the short circuit does not occur at the house junction is … The probability that the short circuit occurs at either the Oven/MW junction or the oven coil is … The probability that both the electronic controls and the thermostat short circuit simultaneously is …

20. EGR 252 - 220 Your turn … Problem 3, page 46

21. EGR 252 - 221 Serial and Parallel Systems For increased safety and reliability, systems are often designed with redundancies. A typical system might look like the following:

22. EGR 252 - 222 Serial and Parallel Systems What is the probability that: –Segment 1 works? – Segment 2 works? –The entire system works? 1 2

23. EGR 252 - 223 Conditional Probability If segment 2 works, what is the probability that component C does not work? 1 2

24. EGR 252 - 224 Conditional Probability If the system works, what is the probability that component D does not work?

25. EGR 252 - 225 Your turn … Problem 3, page 54