10-8

10-8 Counting Principles

Course 3

Warm UpProblem of the DayLesson Presentation

Warm UpAn 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)

11612

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10-8 Counting Principles

Problem of the DayThere 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

VocabularyFundamental Counting Principletree diagramAddition Counting Principle

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10-8 Counting Principles

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10-8 Counting Principles

License plates are being produced that have a single letter followed by three digits. All license plates are equally likely.

Additional Example 1A: Using the Fundamental Counting Principle

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 choices26 • 10 • 10 • 10 = 26,000

The number of possible 1-letter, 3-digit license plates is 26,000.

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Additional Example 1B: Using the Fundamental Counting Principal

Find the probability that a license plate has the letter Q.

1 • 10 • 10 • 1026,000 = 1

26 0.038P(Q ) =

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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 3There are 9 choices for any digit except 3.

P(no 3) = = 0.72926,00018,954

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10-8 Counting Principles

Social Security numbers contain 9 digits. All social security numbers are equally likely.

Check It Out: Example 1A

Find the number of possible Social Security numbers.

Use the Fundamental Counting Principle.Digit 1 2 3 4 5 6 7 8 9Choices 10 10 10 10 10 10 10 10 10

10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 = 1,000,000,000The 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

= = 0.1 101

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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) = ≈ 0.4 1,000,000,000

387,420,489

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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

The different cake possibilities are (yellow, chocolate), (yellow, strawberry), (yellow, vanilla), (white, chocolate), (white, strawberry), and (white, vanilla).

white cake

yellow cake

chocolate icing

vanilla icing

strawberry icing

chocolate icing

vanilla icing

strawberry icing

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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?

None of the lists contains identical items, so use the Addition Counting Principle.

Total ChoicesCourse 3

10-8 Counting Principles

Apples Pears Squash+= +

Apples Pears SquashMacintosh Bosc Acorn

Red Delicious Yellow Bartlett HubbardGold Delicious Red Bartlett

Additional Example 3 Continued

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10-8 Counting Principles

T 3 3 2+= + = 8

There are 8 items to choose from.

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?

None of the lists contains identical items, so use the Addition Counting Principle.

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10-8 Counting Principles

T-Shirts Sweaters PantsLong Sleeve Wool DenimShirt Sleeve Cotton Khaki

Pocket PolyesterCashmere

Additional Example 3 Continued

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10-8 Counting Principles

T 3 4 2+= + = 9

There are 9 items to choose from.

Total Choices T-shirts Sweaters Pants+= +

Lesson Quiz: Part IPersonal 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 containa 6. 0.6561

6,760,000

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10-8 Counting Principles

Lesson Quiz: Part IIA 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?

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

8

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Course 3

10-8 Counting Principles