Probability Review: Sample Space, Basic Probability, Venn Diagrams, Tree Diagrams

1. Unit 7: Probability Final Exam Review

2. Topics to Include • Sample Space • Basic Probability • Venn Diagrams • Tree Diagrams • Fundamental Counting Principle • Permutations • Combinations • Conditional Probability

3. Sample Space • Sample Space is a LISTof all of the possible OUTCOMESin a scenario • Example: Write the sample space for the types of cards that can be selected in a deck of cards • A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K • Example: List the sample space for rolling a dice • 1, 2, 3, 4, 5, 6

4. Sample Space • Now you try: List out the sample space: List the sample space for the colors in a bag of regular M&Ms

5. Basic Probability • Probability is used a lot in a DECK OF CARDS • The DENOMINATORshould always be 52 • Always REDUCE! • Example • P(black card) • P(card <4) • Answer: 26/52 or ½ • Answer: 12/52 or 3/13

6. Basic Probability • You Try: • P(Face Card that is not a King) • P(5 or a 9) • P(Red card or an 8)

7. Venn Diagrams • Venn Diagrams are a VISUALrepresentation used to COMPAREdata • When filling out a Venn Diagram, always STARTin the MIDDLE • Always check to make sure that ALLdata has been used. If not, complete the diagram with a number OUTSIDEof the circles.

8. Venn Diagrams • Fill out the Venn Diagram and then solve the problems that follow: 55 people were questioned at a concert. 43 people said that they like to stand during a concert. 23 people said that they like to sit during a concert. 14 people said that they like to do both. 1. How many people do not like to sit during a concert 2. How many people either like to sit or stand during a concert, but not both? 3. What is the probability that a person likes to only sit at concerts?

9. Tree Diagrams • Tree Diagrams are VISUALrepresentations of the possible OUTCOMESin a scenario • Example: Outcomes for flipping 3 coins

10. Tree Diagrams • Draw a tree diagram to represent the situation At a small ice cream parlor, you can choose from 4 flavors of ice cream, 3 toppings, and 2 syrups. Make a tree diagram to represent the possible choices you can make for an ice cream sundae.

11. Fundamental Counting Principle • The Fundamental Counting Principle is a SHORTCUTto a Tree Diagram. • All you need to do is MULTIPLYthe choices together to find out how many outcomes are possible. • Example: In how many ways can you select one dog, one gorilla, and one penguin from a collection of 7 different dogs, 6 different gorillas, and 3 different penguins? • Answer: 7 6 3 = 126

12. Fundamental Counting Principle • Use the Fundamental Counting Principle to find the number of outcomes for the situation: You want to buy the perfect tree and decorations for the holiday season. You can choose from a douglas fir tree, noble fir tree, cedar tree, or a spruce tree. You can choose from a strand of white lights, colored lights, white lights that twinkle, and colored lights that twinkle. You can choose from striped ornaments, solid ornaments, or handmade ornaments. Lastly, you can choose from 6 different tree toppers. How many ways can you choose to make the perfect tree?

13. Permutations • Permutations are used to find the number of DIFFERENTways to order items • ORDER MATTERS • That means that every time you flip 2 items, you create a NEWorder • To solve Permutations • Use can use BLANKS • Use can use FACTORIALS(!) • You can use nPr in your calculator • n is the number of items you HAVE • r is the number of items you WANTto put in order

14. Permutations • Example How many ways can you arrange the letters in the word “SNOWMAN” to make a new word? • How many ways can you put 4 books in order on a bookshelf from a selection of 10 books? P4 = 5040

15. Permutations • You Try: • How many ways can you rearrange the letters in the word “papajohnspizza” to create a new word? • How many ways can 6 people choose to sit in a row that has 8 empty seats?

16. Combinations • Combinations are used to find the number of OUTCOMESthat can happen in a scenario • ORDER DOES NOT MATTER • That means that even if you pick items in a different order, you still have the SAMEnumber of items • To solve Combinations • Use nCrin your calculator • n is the number of items you HAVE • r is the number of items you WANTto select

17. Combinations • Examples: • How many ways can you select a group of 6 people from a class of 25 people? 25C6 = 177100 • How many ways can you choose 1 math class, 1 english class, and 1 PE class if there are 5 math classes to choose from, 3 english classes to choose from, and 7 PE classes to choose from? 5C1 3C1 7C1 = 105

18. Combinations • You Try: • How many ways can you select 3 toys from a bin of 23 toys? • How many ways can you choose a pizza with 4 toppings if you have 21 toppings to choose from?

19. CONDITIONAL PROBABILITY • Conditional probability typically is presented in a 2-WAY-TABLE • You may have to fill in some of the numbers if they are missing.

20. CONDITIONAL PROBABILITY • You will be asked to find certain probabilities, especially GIVENprobabilities. • Given probabilities are written like this: P(Like Skateboards | Like Snowmobiles) • This is read as “The probability that a person LIKES SKATEBOARDS GIVEN THEY LIKE SNOWMMOBILES • Use this formula: P(A and B) P(B) • So the answer to P(Like Skateboards | Like Snowmobiles) = 80 = 16 105 21

21. CONDITIONAL PROBABILITY • You try: • Fill in the 2-way-table 1. P(Adult) = 2. P(Balcony) = 3. P(Circle | Child) = 4. P(Adult | Balcony) =

22. ALL DONE