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Sample Space, S

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 }

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Sample Space, S

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  1. 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…

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

  3. Events • A subset of the sample space reflecting the specific occurrences of interest. • Example, • All EVE students, V =

  4. 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)

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

  6. Other Venn Diagram Examples • Mutually exclusive events • Subsets

  7. 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. 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 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.

  9. Another Example • Example 2.14, pg. 32

  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 … • In general, the number of permutations of n objects is n!

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

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

  14. 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(A1U A2U … U Ak) = P(A1) + P(A2) + … P(Ak)

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