d2.b how do i apply the fundamental & addition counting principles to find the number of...

D2.b How Do I Apply the Fundamental & Addition Counting Principles To Find The Number of Outcomes?

Course 3

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson 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

Course 3

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

Course 3

10-8 Counting Principles

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

Course 3

10-8 Counting Principles

Vocabulary

Fundamental Counting Principletree diagramAddition Counting Principle

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Course 3

10-8 Counting Principles

Course 3

10-8 Counting Principles

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

Example 1: Using the Fundamental Counting Principle

**Find the number of possible license plates.

Use the Fundamental Counting Principle.

letter first digit second digit third digit

26 choices 10 choices 10 choices 10 choices

26 • 10 • 10 • 10 = 26,000The number of possible 1-letter, 3-digit license plates is 26,000.

Course 3

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 9

Choices 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.

Course 3

10-8 Counting Principles

Example 2: Using the Fundamental Counting Principle

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

Check It Out: Example 2B

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 10

1

Course 3

10-8 Counting Principles

Example 3: Using the Fundamental Counting Principle

Find the probability that a license plate, with a single letter followed by three digits, 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

Course 3

10-8 Counting Principles

Check It Out: Example 3A

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

Course 3

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.

Course 3

10-8 Counting Principles

Example 4: 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 4 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 4A

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.

Course 3

10-8 Counting Principles

Check It Out: Example 4A 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

Course 3

10-8 Counting Principles

Lesson Quiz

A lunch menu consists of 3 types of sandwiches, 2 types of soup, and 3 types of fruit.

1. What is the total number of lunch items on the t menu?

2. 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