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 = {e1, e2, e3, …, en} where ei = the outcomes of interest Note: sometimes a tree diagram is helpful in determining the sample space… other examples: gender of SOE students, S = {M, F} specialization of BSE students in MUSE, S = {BME, CPE, ECE, EVE, ISE, MAE}

Sample Space Example: The sample space of gender and specialization of all BSE students in the School of Engineering is … S = {BMEF, BMEM, CPEF, CPEM, ECEF, ECEM, EVEF, EVEM, ISEF, ISEM, MAEF, MAEM}, SEE DIAGRAM …

Events A subset of the sample space reflecting the specific occurrences of interest. Example, All EVE students, V = V = {EVEF, EVEM} other examples, all female students: F = {BMEF, CPEF, ECEF, EVEF, ISEF, MAEF} all male MAE students, MM = {MAEM}

Events Complement of an event, (A’, if A is the event) e.g., students who are not EVE, 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) V’ = {BMEF, BMEM, CPEF, CPEM, ECEF, ECEM, ISEF, ISEM, MAEF, MAEM} V int. F = {EVEF} mutually exclusive example, students who are EVE and male MAE. Union example, students who are EVE OR students who are male MAE

Venn Diagrams Example, events V (EVE students) and F (female students)

Other Venn Diagram Examples Mutually exclusive events Subsets mutually exclusive example, EVE and MAE students subset example, female students and female ISE students

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.

Sample Points Multiplication Rule If event A can occur n1 ways and event B can occur n2 ways, then an event C that includes both A and B can occur n1 n2 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.

Another Example Example 2.14, pg. 32

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! First position has 6 options, second has 5, 3rd has 4, etc., so 6*5*4*3*2*1 = 720 by definition, 1! = 1 and 0! = 1.

Permutations If we take the number of specializations 3 at a time (n = 6, r = 3), the number of permutations is In general,

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? 12P3 = 12!/(12-3)! = 479,001,600/362,880 = 1320 or, 12*11*10 = 1320

Combinations Selections of subsets without regard to order. Example: How many ways can we select 3 guides from the 12 Ambassadors? Note: difference between permutations and combinations Perm. = “how many ways can I get an Ambassador given I have 12 choices to start, then 11, then 10?” Comb. = “If I’m going to divide the Ambassadors into groups of 4, how many different groups can I have?” (12 choose 3) = 12!/3!(12-3)! = 479,001,600/(6*362,880) = 220

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(A1 U A2 U … U Ak) = P(A1) + P(A2) + … P(Ak)