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									10-8 Counting Principles

Warm Up Problem of the Day Lesson Presentation

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10-8 Counting Principles

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) 1 2. P(rolling a prime number) 1 3. P(rolling a number > 7)
6 1 2

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10-8 Counting Principles

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

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10-8 Counting Principles

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

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10-8 Counting Principles Here Insert Lesson Title

Vocabulary
Fundamental Counting Principle tree diagram Addition Counting Principle

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10-8 Counting Principles

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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 first digit second digit third digit 10 choices

26 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 1 ) = 1 • 10 • 10 • 10 =  0.038 26,000 26

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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) = 18,954 = 0.729 26,000
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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
10

3

4

5
10

6

7

8
10

9
10

Choices 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

=

1 = 0.1 10

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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) = 387,420,489 1,000,000,000 ≈ 0.4

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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 vanilla icing chocolate icing strawberry icing white cake vanilla icing chocolate 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 Macintosh Red Delicious Gold Delicious Pears Bosc Yellow Bartlett Red Bartlett Squash Acorn Hubbard

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

+

2

=8

There are 8 items to choose from.

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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 Long Sleeve Shirt Sleeve Pocket Sweaters Wool Cotton Polyester Cashmere Pants Denim Khaki

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-shirts + Sweaters +

Pants

T

=

3

+

4

+

2

=9

There are 9 items to choose from.

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10-8 Counting Principles Here Insert Lesson Title
Lesson Quiz: Part I 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. 6,760,000 2. Find the probability that a PIN does not contain 0.6561 a 6.

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Lesson Quiz: Part II A lunch menu consists of 3 types of sandwiches, 2 types of soup, and 3 types of fruit. 3. What is the total number of lunch items on the t menu? 8 4. A student wants to order one sandwich, one t bowl of soup, and one piece of fruit. How many t different lunches are possible? 18

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