2. The Counting Principle

3. Counting Outcomes Have you ever seen or heard the Subway or Starbucks advertising campaigns where they talk about the 10,000 different combinations of ways to order a sub or drink?

4. Counting Outcomes Have you ever seen or heard the Subway or Starbucks advertising campaigns where they talk about the 10,000 different combinations of ways to order a sub or drink? When companies like these make these claims they are using all the different condiments and ways to serve a drink.

5. Counting Outcomes - These companies can use (2) ideas related to combinations to make these claims: (1) TREE DIAGRAMS (2) THE FUNDAMENTAL COUNTING PRINCIPLE

6. Counting Outcomes (1) TREE DIAGRAMS A tree diagram is a diagram used to show the total number of possible outcomes in a probability experiment.

7. Counting Outcomes (2) THE COUNTING PRINCIPLE The Counting Principle uses multiplication of the number of ways each event in an experiment can occur to find the number of possible outcomes in a sample space. http://www.youtube.com/watch?v=8WdSJh EIrQk&safe=active http://www.youtube.com/watch?v=8WdSJh EIrQk&safe=active

8. Counting Outcomes Example 1: Tree Diagrams. A new polo shirt is released in 4 different colors and 5 different sizes. How many different color and size combinations are available to the public? Colors – (Red, Blue, Green, Yellow) Styles – (S, M, L, XL, XXL)

9. The Tree Diagram

10. A Different Way Example 1: The Counting Principle. A new polo shirt is released in 4 different colors and 5 different sizes. How many different color and size combinations are available to the public? Colors – (Red, Blue, Green, Yellow) Styles – (S, M, L, XL, XXL)

11. Counting Outcomes Example 1: The Fundamental Counting Principle. Answer. Number of Number of Number of Possible Styles Possible Sizes Possible Comb. 4 x 5 =20

12. Counting Outcomes  Tree Diagrams and The Fundamental Counting Principle are two different algorithms for finding sample space of a probability problem.  However, tree diagrams work better for some problems and the fundamental counting principle works better for other problems.

13. So when should I use a tree diagram or the fundamental counting principle? - A tree diagram is used to: (1) show sample space; (2) count the number of preferred outcomes. - The fundamental counting principle can be used to: (1) count the total number of outcomes.

14. Counting Outcomes Example 2: Tree Diagram. Tamara spins a spinner two times. What is her probability of spinning a green on the first spin and a blue on the second spin? You use a tree diagram because you want a specific outcome … not the TOTAL number of outcomes.

15. Counting Outcomes Example 2: Tree Diagram. Tamara spins a spinner two times. What is her probability of spinning a green on the first spin and a blue on the second spin?

16. Counting Outcomes Example 3: The Counting Principle. If a lottery game is made up of three digits from 0 to 9, what is the total number of outcomes? You use the Counting Principle because you want the total number of outcomes. How many possible digits are from 0 to 9?

17. Counting Outcomes Example 3: The Fundamental Counting Principle. If a lottery game is made up of three digits from 0 to 9, what is the total number of possible outcomes? # of Possible # of Possible # of Possible # of Possible Digits Digits Digits Outcomes Digits Digits Digits Outcomes 10 x 10 x 10 = 1000 10 x 10 x 10 = 1000 What chance would you have to win if you played one time?

18. Guided Practice: Tree or Counting Principle? (1) How many outfits are possible from a pair of jean or khaki shorts and a choice of yellow, white, or blue shirt? (2) Scott has 5 shirts, 3 pairs of pants, and 4 pairs of socks. How many different outfits can Scott choose with a shirt, pair of pants, and pair of socks?

19. Example 1 You are purchasing a new car. Using the following manufacturers, car sizes and colors, how many different ways can you select one manufacturer, one car size and one color? You are purchasing a new car. Using the following manufacturers, car sizes and colors, how many different ways can you select one manufacturer, one car size and one color? Manufacturer: Ford, GM, Chrysler Car size: small, medium Color: white(W), red(R), black(B), green(G)

20. Solution There are three choices of manufacturer, two choices of car sizes, and four colors. So, the number of ways to select one manufacturer, one car size and one color is: There are three choices of manufacturer, two choices of car sizes, and four colors. So, the number of ways to select one manufacturer, one car size and one color is: 3 ●2●4 = 24 ways.

21. Ex. 2 Using the Fundamental Counting Principle The access code for a car’s security system consists of four digits. Each digit can be 0 through 9. How many access codes are possible if each digit can be repeated? The access code for a car’s security system consists of four digits. Each digit can be 0 through 9. How many access codes are possible if each digit can be repeated?