Sample Spaces, Venn Diagrams, Tree Diagrams and Two-Way Tables CCM2 Unit 6: Probability

Sample Space Sample Space: The set of all possible outcomes of an experiment. List the sample space, S, for each of the following: a. Tossing a coin S = {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}

Unions of Sets The union of two sets (A OR B) is the set of all elements in set A OR set B (or both). This can be written as 𝐴∪𝐵 Example: Given the following sets, A ∪ B A = {1,3,5,7,9,11,13,15} B = {0,3,6,9,12,15} Stress that intersection means AND and that union means OR A ∪ B = {0,1,3,5,6,7,9,11,12,13,15}

Intersections of Sets A ∩ B = {3,9,15} The intersection of two sets (A AND B) is the set of all elements in both set A AND set B. This can be written as A∩𝐵. Example: Given the following sets, find A∩𝐵 A = {1,3,5,7,9,11,13,15} B = {0,3,6,9,12,15} Stress that intersection means AND and that union means OR A ∩ B = {3,9,15}

In a class of 60 students, 21 sign up for chorus, 29 sign up for band, and 5 take both. 15 students in the class are not enrolled in either band or chorus. 6. 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. 7. What is A ∪ B? 8. What is A ∩ B?

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 or A’ 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? AC = {-3,-1,1,3,5,…} Stress that compliment means NOT

Venn Diagrams One way to represent or visualize sets is to use Venn diagrams:

Universe or Universal Set Let U be the set of all students enrolled in classes this semester. U

Let M be the set of all students enrolled in Math this semester. Let E be the set of all students enrolled in English this semester. U E M

Complement of a set Let C be the set of all students enrolled in classes this semester, but who are not enrolled in Math or English U E M C

Intersection () E  M = the set of students in Math AND English U M E

Union () E  M = the set of students in Math OR English U M E

Union of Sets The union of two sets A and B, written , is the set of all members that are common to both sets. is read “A union B” A B A B

Use the Venn Diagram 4. What is A ∩ B? {2, 4} 5. What is A ∪ B? 1 3 2 4 5 6 8 10 What are the elements of set A? {1, 2, 3, 4, 5} What are the elements of set B? {2, 4, 6, 8, 10} 3. Why are 2 and 4 in both sets? 4. What is A ∩ B? {2, 4} 5. What is A ∪ B? {1, 2, 3, 4, 5, 6, 8,10}

A survey of used car salesmen revealed the following information: 24 wear white patent-leather shoes 28 wear plaid trousers 20 wear both of these things 2 wear neither of these things Create a Venn diagram.

2 8 4 Don’t forget these! White Shoes Plaid trousers 20 Always start with the intersection! 8 There should be 28 total in the circle, so subtract here! 4 There should be 24 total in the circle, so subtract here! Explain to students that the intersection is the overlap of the circles and the union is everything in both circles.

2 26 2 + 4 + 20 + 8 = 34 How many salesmen didn’t wear plaid trousers How many salesmen were surveyed? 2 + 4 + 20 + 8 = 34 How many salesmen wore both plaid trousers and white shoes? 2 How many salesmen didn’t wear plaid trousers 26

Two-Way Table: categorical data organized in 2 dimensions

Android vs. iPhone -On ONE post-it, please write your name (first and last). -Stick it to the appropriate quadrant on the whiteboard. Discuss graphical representations. Histogram? No (this is categorical data only) Box plot? (no, we need 2x2 format)

Android vs. iPhone Android iPhone Total Female Male On the board: Post-it with: Male/Female, Date/No Date

Using the class data, answer & discuss: How many students total from this class prefer the Android? How many students prefer the iPhone? How many students are females? How many students are males? How many students are females AND prefer the Android? How many students are males AND prefer the iPhone? These are frequency questions.

For example – a fair coin is spun twice 1st 2nd H HH Possible Outcomes H T HT H TH T T TT

Now you try one! Let’s make a tree diagram to show the different combinations you could make if you could select between white and chocolate milk and either chocolate chip or oatmeal cookies.