1. Lesson 12.4, For use with pages 670-674
1. Find the probability of rolling a number
less than 5 on a dice.
2. Find the probability of rolling a number
less than 7 on a dice.

2. Lesson 12.4, For use with pages 670-674
1. Find the probability of rolling a number
less than 5 on a dice.
3
2ANSWER
2. Find the probability of rolling a number
less than 7 on a dice.
0ANSWER

3. Tree Diagrams and
the Counting Principle
Section 12.4

4. Essential Questions
• What are the differences between
permutations and combinations?
• What are the differences between odds
and probability?
• How is probability used to make
predictions?
• What are the differences between
experimental and theoretical probabilities?

5. • Today we will be learning about different
ways to “count” difficult situations. The
two methods we will use will be
1) Tree Diagrams
2) The Counting Principle

6. Vocabulary
• Tree diagram: a branching diagram that
shows all the possible choices or
outcomes of a process carried out in
several stages.

7. • At Sam’s Deli, you can make a sub
sandwich using the following:
Breads: wheat, rye
Meat: ham, turkey, salami
How many ways can you put a sandwich
together? Use a tree diagram

8. Breads: wheat, rye
Meat: ham, turkey, salami
Wheat Rye
H T S H T S

9. Breads: wheat, rye
Meat: ham, turkey, salami
Now if we added cheese to the sandwich.
Cheese: cheddar, American, Swiss
Wheat Rye
H T S H T S
C A S C A S C A S C A S C A S C A S

10. EXAMPLE 1 Making a Tree Diagram
Make a tree diagram to help you choose an outfit.
You can choose a T-shirt (T), button-down shirt (B), sweater
(S), or a polo (P) as a top, and jeans (J), khakis (K), or dress
pants (D) for pants. How many different outfits are possible?
There are 12 different possible outfits.

11. • The “Counting Principle” states
“if a situation can occur in m ways, and a
second situation can occur in n ways, then
these things can occur together in
“m x n” ways.

12. EXAMPLE 2 Using the Counting Principle
Skateboards
ANSWER
You can build 15 different skateboards.
5 3 = 15
decks wheel
assemblies
To build a skateboard, you can
choose one deck and one type of
wheel assembly from those
shown. To count the number of
different skateboards you can
build, use the counting principle.

13. • How many different results could you get
by spinning both spinners below?
– 4 x 3 = 12
1 2
3
4
Blue
Green
Red

14. EXAMPLE 3 Using the Counting Principle
Passwords
You are choosing a password that starts with 3 letters
and then has 2 digits. How many different passwords
are possible?
SOLUTION
26 26 26 10 10 =
letters digits
ANSWER
There are 1,757,600 different possible passwords.
1,757,600

15. EXAMPLE 3 Using the Counting Principle
Passwords
You are choosing a password that starts with 3 letters
and then has 2 digits. How many different passwords
are possible?
26 26 26 10 10 =
letters digits
1,757,600
How would the number of possible passwords
change if you were to add one letter to the
requirement?
26 x 26 x 26 x 26 x 10 x 10 = 45,697,600

16. GUIDED PRACTICE for Example 1, 2, and 3
2. Soccer Uniforms
Your soccer team’s uniform choices include yellow
and green shirts, white, black, and green shorts,
and four colors of socks.
How many different uniforms are possible?
Soccer team have 2 shirts 3 shorts and 4 socks to
select. Use the counting principle.
ANSWER
The number of possible uniform are 2 3 4 = 24
uniform

17. Probability
Number of favorable outcomes
Number of possible outcomes

18. EXAMPLE 4 Solve a Multi-Step Problem
Number Cubes
You and three friends each roll a number cube.
What is the probability that you each roll the same
number?
STEP 1 List the favorable outcomes. There are 6:
Everyone could roll 1-1-1-1, 2-2-2-2, 3-3-3-3,
4-4-4-4, 5-5-5-5, or 6-6-6-6.
STEP 2 Find the number of possible outcomes using
the counting principle.
6 6 6 6 = 1296
4 number cubes

19. EXAMPLE 4 Solve a Multi-Step Problem
STEP 3 Find the probability:
Number of favorable outcomes
Number of possible outcomes
= 1296
6
= 216
1
ANSWER
The probability that you each roll the same number
is .
216
1

20. Homework
• Page 672 #1-12