1. Counting Outcomes

2. Counting Outcomes Objectives:
(1) To develop fluency with counting strategies to determine the sample space for an event.
Essential Questions:
(1) How can I construct and use a frequency table (a.k.a. tree diagram)?
(2) How can I use the Fundamental Counting Principle to find the number of outcomes?

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 FUNDAMENTAL COUNTING PRINCIPLE
The Fundamental 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.

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. Counting Outcomes Example 1: Tree Diagrams. Answer.
Red Blue Green Yellow
S M L XL XXL S M L XL XXL
S M L XL XXL S M L XL XXL
There are 20 different combinations.

10. Counting Outcomes Example 1: The Fundamental 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 = 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. 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?

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?
Green Blue
Green Blue Green Blue
Only one outcome has green then blue, and there are 4 possibilities…so the P(green, blue) = ¼ or .25 or 25%

15. 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 probability of winning the game?

16. 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 probability of winning if you buy 1 ticket?
# of Possible # of Possible # of Possible # of Possible
Digits Digits Digits Outcomes
x x =
Because there are 1000 different possibilities, buying one ticket gives you a 1/1000 probability or or 0.1% chance of winning.

17. Counting Outcomes
Guided Practice: Determine the probability for each problem.
(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?

18. Counting Outcomes
Guided Practice: Determine the probability for each problem.
(1) Jean Shorts Khaki Shorts
Yellow White Blue Yellow White Blue
JSYS JSWS2 JSBS3 KSYS4 KSWS5 KSBS6
(2) Number Number Number Number
Of Shirts Of Pants Of Socks Of Outfits
5 x x =

19. Counting Outcomes
Real World Example: The Fundamental Counting Principle.
How many seven digit telephone numbers can be made up using the digits 0-9, without repetition?

20. Counting Outcomes
Real World Example: The Fundamental Counting Principle.
How many seven digit telephone numbers can be made up using the digits 0-9, without repetition?
Answer: 604,800 different numbers

21. Counting Outcomes Real World Example: Tree Diagram.
Kaitlyn tosses a coin 3 times. Draw a picture showing the possible outcomes. What is the probability of getting at least 2 tails?

22. Counting Outcomes Real World Example: Tree Diagram.
Kaitlyn tosses a coin 3 times. Draw a picture showing the possible outcomes. What is the probability of getting at least 2 tails?
Answer: P(at least 2 tails) = ½

23. Counting Outcomes Summary:
- A tree diagram is used to show all of the possible outcomes, or sample space, in a probability experiment.
- The fundamental counting principle can be used to count the number of possible outcomes given an event that can happen in some number of ways followed by another event that can happen in some number of different ways.

24. Counting Outcomes
Summary: 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.

25. Counting Outcomes
Homework: -