1. Probability Probability II

2. Opening Routine # 1

4. 4 Venn diagrams represent events as circles enclosed in a rectangle. The rectangle represents the sample space and each circle represents an event.

5. 5

6. 6 If events E and F have no simple events in common or cannot occur simultaneously, they are said to be disjoint or mutually exclusive.

7. 7 Events E and F are Mutually Exclusive Events E, F and G are Mutually Exclusive

8. Begin the Venn diagram by assigning labels. One circle will be labeled music, the other drama, and the overall diagram can be called anything you want. Let’s call this diagram Elective. Since there are 265 students taking music only, write 265 in the music circle. This number won’t be in the overlapping portion because these students aren’t taking drama. The number of students taking both music and drama is 342. So write this number where the circles overlap. Write 293 in the drama circle for the 293 students taking drama only.

10. Let Set E = the universal set {“sample space”} (all the students in the school) Set E = 990 students Let Set A = all music students Set A = 607 students (265 + 342 = 607!) Let Set B = all drama students Set B = 635 students (293 + 342 = 635!)

13. S AB Event Relations The intersection of two events, A and B, is the event that both A and B occur when the experiment is performed. We write A  B. If two events A and B are mutually exclusive, then P(A  B) = 0.

15. S Event Relations The beauty of using events, rather than simple events, is that we can combine events to make other events using logical operations: and, or and not. union or both The union of two events, A and B, is the event that either A or B or both occur when the experiment is performed. We write A  B AB

17. S Event Relations The complement of an event A consists of all outcomes of the experiment that do not result in event A. We write A ’. A A’A’

19. Complementary Events The complement of event E is the set of all outcomes in a sample space that are not included in event E. The complement of event E is denoted by Properties of Probability:

21. What is D’ ? Recall:

22. Unions, Intersections, and Complements A die is rolled. Let A be the event “Number rolled is even” and B be the event “Number rolled is at least 4.” Then A = [2, 4, 6] andB = [4, 5, 6]

23. Example Select a student from the classroom and record his/her hair color and gender. A: student has brown hair B: student is female C: student is male What is the relationship between events B and C?: A ’ : B  C: B  C: Mutually exclusive; B = C ’ Student does not have brown hair Student is both male and female =  Student is either male and female = all students = S