Sample Spaces, Subsets and Basic Probability Assignment 2: Guided Notes

Warm-Up 1 If you have a standard dice what are the possibilities that you could roll? If you flip a coin, what are the possibilities that you could flip? Simplify:

Sample Space Definition: A Sample Space is the set of all possible outcomes of an experiment. Example: The sample space, S, of the days of the school week is S = {Monday, Tuesday, Wednesday, Thursday, Friday}

Example 1 List the sample space, S, for each of the following: Tossing a coin Rolling a six-sided die Drawing a marble from a bag that contains two red, three blue and one white marble

Example 1: Answers List the sample space, S, for each of the following: a. Tossing a coin S = {Heads, Tails} or {H, T} b. Rolling a six-sided die S = {1,2,3,4,5,6} c. Drawing a marble from a bag that contains two red, three blue and one white marble S = {red, red, blue, blue, blue, white}

Intersections and Unions of Sets The intersection of two sets (denoted A  B) is the set of all elements in both set A AND set B. (must be in both) The union of two sets (denoted A  B) is the set of all elements in either set A OR set B. (everything) Stress that intersection means AND and that union means OR

Ex 1 Example: Given the following sets, find A  B and A  B A  B = {4, 7} (because 4 and 7 are the only number in BOTH) A  B = {1, 2, 3, 4, 5, 6, 7, 8, 9}

Example 2 Given the following sets, find A  B and A  B Stress that intersection means AND and that union means OR

Venn Diagrams Sometimes drawing a diagram helps in finding intersections and unions of sets. A Venn Diagram is a visual representation of sets and their relationships to each other using overlapping circles. Each circle represents a different set.

Example 3 What are the elements of set A? Factors of 12 Factors of 16 3 6 12 1 2 4 8 16 What are the elements of set A? What are the elements of set B? Why are 1, 2, and 4 in both sets? What is A  B? What is A  B?

Example 3: Answers What are the elements of set A? {1,2,3,4,6,12} Factors of 12 Factors of 16 3 6 12 1 2 4 8 16 What are the elements of set A? {1,2,3,4,6,12} What are the elements of set B? {1,2,4,8,16} Why are 1, 2, and 4 in both sets? 1,2 and 4 are factors of both 12 and 16

Example 3: Answers What is A  B? {1,2,4} What is A  B? Factors of 12 Factors of 16 3 6 12 1 2 4 8 16 What is A  B? {1,2,4} What is A  B? {1,2,3,4,6,8,12,16}

Example 4 In a freshman class of 60 students, 21 sign up for chorus, 29 sign up for band, and of those, 5 take both. 15 students in the class are not enrolled in either band or chorus. Put this information into a Venn Diagram. If the sample space, S, is the set of all students in the class, let students in chorus be set A and students in band be set B. What is A  B? What is A  B?

S. A. B.

16 5 Example 4: Answers A  B = {45} A  B = {5} 24 S. Students in the class 15 A. Students in Chorus 16 5 B. Students in Band 24 A  B = {45} A  B = {5}

Complement of a set The Complement of a set is the set of all elements NOT in the set. The complement of a set, A, is denoted as AC Ex: S = {…-3,-2,-1,0,1,2,3,4,…} A = {…-2,0,2,4,…} If A is a subset of S, what is AC? Stress that compliment means NOT

Complement of a set The Complement of a set is the set of all elements NOT in the set. The complement of a set, A, is denoted as AC Ex: S = {…-3,-2,-1,0,1,2,3,4,…} (all integers) A = {…-2,0,2,4,…} (even numbers) If A is a subset of S, what is AC? AC = {-3,-1,1,3,5,…} (odds) Stress that compliment means NOT

16 5 Example 5: ANSWERS What is AC? 39 BC? 31 What is (A  B)C? 55 S. Students in the class 15 A. Students in Chorus 16 5 B. Students in Band 24 What is AC? 39 BC? 31 What is (A  B)C? 55 What is (A  B)C? 15

Practice