the fundamental counting principle 10-6 learn to find the number of possible outcomes in an...

The Fundamental Counting Principle10-6

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


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

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.


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 ) =


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


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

Example 2A

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.


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


Example 2C

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


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.


Example 3: 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.


Example 3 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).


Example 4

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.


Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems


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.

2. Find the probability that a PIN does not containa 6. 0.6561

6,760,000


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?

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


1. A login password contains 3 letters followed by 2 digits. Identify the number of possible login passwords.

A. 175,760

B. 676,000

C. 1,757,600

D. 6,760,000



2. Employee identification codes at a company contain 2 letters followed by 4 digits. Assume that all codes are equally likely. Identify the probability that an ID code does not contain the letter I.

A. 0.6567

B. 0.7493

C. 0.8321

D. 0.9246



3. A restaurant offers 4 main courses, 3 desserts, and 5 types of juices. What is the total number of items on the menu?

A. 3

B. 7

C. 9

D. 12



4. A restaurant offers 3 types of starters, 4 types of sandwiches, and 4 types of salads for dinner. Visitors select one starter, one sandwich, and one salad. How many different dinners are possible?

A. 3

B. 4

C. 11

D. 48
