1. 9-5 The Fundamental Counting Principle Warm Up Problem of the Day
Lesson Presentation
Pre-Algebra

2. 9-5 The Fundamental Counting Principle Warm Up
Pre-Algebra
9-5
The Fundamental Counting Principle
Warm Up
An experiment consists of rolling a fair number cube with faces numbered 2, 4, 6, 8, 10, and 12. Find each probability.
1. P(rolling an even number)
2. P(rolling a prime number)
3. P(rolling a number > 7) 1 1 6 1 2

3. Problem of the Day
There are 10 players in a chess tournament. How many games are needed for each player to play every other player one time? 45

4. Learn to find the number of possible outcomes in an experiment.

5. Vocabulary
Fundamental Counting Principal
tree diagram

7. Additional Example 1A: Using the Fundamental Counting Principal
License plates are being produced that have a single letter followed by three digits. All license plates are equally likely.
A. Find the number of possible license plates.
Use the Fundamental Counting Principal.
letter
first digit
second digit
third digit
26 choices
10 choices
10 choices
10 choices
26  10  10  10 = 26,000
The number of possible 1-letter, 3-digit license plates is 26,000.

8. Additional Example 1B: Using the Fundamental Counting Principal
B. Find the probability that a license plate has the letter Q.
1 26 
1  10  10  10
26,000 =
P(Q ) =
0.038

9. Additional Example 1C: Using the Fundamental Counting Principle
C. Find the probability that a license plate does not contain a 3.
First use the Fundamental Counting Principle to find the number of license plates that do not contain a 3.
26  9  9  9 = 18,954 possible license plates without a 3
There are 9 choices for any digit except 3.
P(no 3) = = 0.729
26,000
18,954

10. Try This: Example 1
Social Security numbers contain 9 digits. All social security numbers are equally likely.
A. Find the number of possible Social Security numbers.
Use the Fundamental Counting Principal.
Digit 1 2 3 4 5 6 7 8 9
Choices 10
10  10  10  10  10  10  10  10  10 = 10,000,000,000
The number of Social Security numbers is 10,000,000,000.

11. B. Find the probability that the Social Security number contains a 7.
Try This: Example 1B
B. Find the probability that the Social Security number contains a 7.
P(7 _ _ _ _ _ _ _ _) = 1 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10
10,000,000,000
= = 0.01
100 1

12. Try This: Example 1C
C. Find the probability that a Social Security number does not contain a 7.
First use the Fundamental Counting Principle to find the number of Social Security numbers that do not contain a 7.
P(no 7 _ _ _ _ _ _ _ _) = 9 • 9 • 9 • 9 • 9 • 9 • 9 • 9 • 9
10,000,000,000
P(no 7) = ≈ 0.04
10,000,000,000
387,420,489

13. The Fundamental Counting Principle tells you only the number of outcomes in some experiments, not what the outcomes are. A tree diagram is a way to show all of the possible outcomes.

14. Additional Example 2: Using a Tree Diagram
You have a photo that you want to mat and frame. You can choose from a blue, purple, red, or green mat and a metal or wood frame. Describe all of the ways you could frame this photo with one mat and one frame.
You can find all of the possible outcomes by making a tree diagram.
There should be 4  2 = 8 different ways to frame the photo.

15. Additional Example 2 Continued
Each “branch” of the tree diagram represents a different way to frame the photo. The ways shown in the branches could be written as (blue, metal), (blue, wood), (purple, metal), (purple, wood), (red, metal), (red, wood), (green, metal), and (green, wood).

16. Try This: Example 2
A baker can make yellow or white cakes with a choice of chocolate, strawberry, or vanilla icing. Describe all of the possible combinations of cakes.
You can find all of the possible outcomes by making a tree diagram.
There should be 2  3 = 6 different cakes available.

17. Try This: Example 2
yellow cake
The different cake possibilities are (yellow, chocolate), (yellow, strawberry), (yellow, vanilla), (white, chocolate), (white, strawberry), and (white, vanilla).
vanilla icing
chocolate icing
strawberry icing
white cake
vanilla icing
chocolate icing
strawberry icing

18. Lesson Quiz
Personal identification numbers (PINs) contain 2 letters followed by 4 digits. Assume that all codes are equally likely.
1. Find the number of possible PINs.
2. Find the probability that a PIN does not contain a 6.
3. For lunch a student can choose one sandwich, one bowl of soup, and one piece of fruit. The choices include grilled cheese, peanut butter, or turkey sandwich, chicken soup or clam chowder, and an apple, banana, or orange. How many different lunches are possible?