Transcript Course 3 10-8

10-8 Counting Principles
Learn to find the number of possible outcomes in an
experiment.
Course 3
Principles
10-8 Counting
Insert Lesson
Title Here
Vocabulary
Fundamental Counting Principle
tree diagram
Addition Counting Principle
Course 3
10-8 Counting Principles
Course 3
10-8 Counting Principles
Additional Example 1A: Using the Fundamental
Counting Principle
License plates are being produced that have a single letter
followed by three digits. All license plates are equally likely.
Find the number of possible license plates.
Use the Fundamental Counting Principal.
letter
26 choices
first digit
second digit
third digit
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.
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10-8 Counting Principles
Additional Example 1B: Using the Fundamental
Counting Principal
Find the probability that a license plate has the letter Q.
P(Q
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)=
1 • 10 • 10 • 10
26,000
=
1
 0.038
26
10-8 Counting Principles
Additional Example 1C: Using the Fundamental
Counting Principle
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) =
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18,954
= 0.729
26,000
10-8 Counting Principles
Check It Out: Example 1A
Social Security numbers contain 9 digits. All social security
numbers are equally likely.
Find the number of possible Social Security numbers.
Use the Fundamental Counting Principle.
Digit
1
2
3
4
5
6
7
8
9
Choices 10 10 10 10 10 10 10 10 10
10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 = 1,000,000,000
The number of Social Security numbers is 1,000,000,000.
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10-8 Counting Principles
Check It Out: Example 1B
Find the probability that the Social Security number contains a 7.
P(7 _ _ _ _ _ _ _ _) = 1 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10
1,000,000,000
=
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1= 0.1
10
10-8 Counting Principles
Check It Out: Example 1C
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
1,000,000,000
P(no 7) =
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387,420,489
≈ 0.4
1,000,000,000
10-8 Counting Principles
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.
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10-8 Counting Principles
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.
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10-8 Counting Principles
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).
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10-8 Counting Principles
Check It Out: 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.
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10-8 Counting Principles
Check It Out: Example 2 Continued
yellow cake
chocolate icing
vanilla icing
strawberry
icing
white cake
chocolate icing
vanilla icing
strawberry
icing
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The different cake
possibilities are (yellow,
chocolate), (yellow,
strawberry), (yellow,
vanilla), (white, chocolate),
(white, strawberry), and
(white, vanilla).
10-8 Counting Principles
Additional Example 3: Using the Addition Counting
Principle
The table shows the items available at a farm stand. How
many items can you choose from the farm stand?
Apples
Pears
Squash
Macintosh
Bosc
Acorn
Red Delicious
Yellow Bartlett
Hubbard
Gold Delicious
Red Bartlett
None of the lists contains identical items, so use the Addition
Counting Principle.
Total Choices
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=
Apples
+
Pears
+
Squash
10-8 Counting Principles
Additional Example 3 Continued
T
=
3
+
3
There are 8 items to choose from.
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+
2
=8
10-8 Counting Principles
Check It Out: Example 3
The table shows the items available at a clothing store. How
many items can you choose from the clothing store?
T-Shirts
Sweaters
Pants
Long Sleeve
Wool
Denim
Shirt Sleeve
Cotton
Khaki
Pocket
Polyester
Cashmere
None of the lists contains identical items, so use the Addition
Counting Principle.
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10-8 Counting Principles
Additional Example 3 Continued
Total Choices
T
=
=
T-shirts +
3
+
4
There are 9 items to choose from.
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Sweaters +
+
2
Pants
=9