4.4 Counting Rules - WordPress.com€¦ · A Question to Ponder: A box contains 3 red chips, 2 blue chips and 5 green chips. A chip is selected, and not replaced and a second chip

Sec 4.4

Counting Rules

Bluman, Chapter 4

A Question to Ponder:

A box contains 3 red chips, 2 blue chips and 5 green chips. A chip is selected, replaced and a second chip is selected.

Display the sample space.

Do you think each event is

equally likely?


A box contains 3 red chips, 2 blue chips and 5 green chips. A chip is selected, replaced and a second chip is selected. Find the following probabilities

a) Selecting 1 red chip and then 1 blue.

b) Selecting 1 blue chip and then 1 green chip.

c) Selecting 2 blue chips.


A box contains 3 red chips, 2 blue chips and 5 green chips. A chip is selected, and not replaced and a second chip is selected.

Display the sample space

Do you think each event is equally

likely?


A box contains 3 red chips, 2 blue chips and 5 green chips. A chip is selected, and not replaced and a second chip is selected. Find the probability of each event

a) Selecting 1 red chip and then 1 blue chip.

b) Selecting 1 blue chip and then 1 green chip.

c) Matching chips.

Another Question to Ponder

A game is played by drawing four cards from

an ordinary deck and replacing each card

after it is drawn. Find the probability of

winning if at least one ace is drawn.

The fundamental counting rule is also

called the multiplication of choices.

In a sequence of n events in which the

first one has k1 possibilities and the second

event has k2 and the third has k3, and so

forth, the total number of possibilities of the

sequence will be

k1 · k2 · k3 · · · kn

4.4 Counting Rules

Bluman, Chapter 4

Chapter 4

Probability and Counting Rules

Section 4-4

Example 4-39

Page #225

8 Bluman, Chapter 4

Example 4-39: Paint Colors A paint manufacturer wishes to manufacture several

different paints. The categories include

Color: red, blue, white, black, green, brown, yellow

Type: latex, oil

Texture: flat, semigloss, high gloss

Use: outdoor, indoor

How many different kinds of paint can be made if you can

select one color, one type, one texture, and one use?

9 Bluman, Chapter 4

# of # of # of # ofcolors types textures uses

7 2 3 2

84 different kinds of paint

Example 4-40

There are four blood types, A, B, AB, and

O. Blood can also be Rh+ and Rh-.

Finally, a blood donor can be classified as

either male or female. How many different

ways can a donor have his or her blood

labeled?

Blood type ∙sign ∙gender

4∙2∙2=16

Example 4-41

The digits 1, 2, 3, 4, 5 and 6 are to be

used in a four-digit ID card. How many

different cards are possible if repetitions

are permitted?

First digit ∙ second digit ∙ third ∙ fourth

6 ∙ 6 ∙ 6 ∙ 6= 1296

Counting Rules

Factorial is the product of all the positive

numbers from 1 to a number.

Bluman, Chapter 4

! 1 2 3 2 1

0! 1

n n n n

While you wait:

In how many ways can 3 people stand in

line at a store?

In how many ways can 4 people be

chosen from a class of 12 to go on a FT?

Bluman, Chapter 4

Review topics

In how many ways can two items be chosen

from a selection of 4 . No replacement.

Bluman, Chapter 4

Consider the following cases

Ten runners compete in

an event.

The top 3 finishers

advance to the State

level.

Ten runners compete in

an event.

The first to cross the

finish line will receive a

gold medal, the second

place a silver and finally

the third finisher a

bronze medal.

Bluman, Chapter 4

Bluman, Chapter 4

There is 5-question

True or False test:

T=true; F=false

Are the following the

same result:

TFFTT OR FFTTT

There are 5 DVDs.

Marge is sick and

staying home. In how

many ways can Marge

watch the DVDs.

Bluman, Chapter 4

Bluman, Chapter 4

Marge is shopping at

the local grocery store

and needs five items

from the produce

department. Jack is

also buying the same

five items.

Do you think Jack will

make his choices in the

same order as Marge?

Does it matter at the

end how they make

their choices?

There are 8 DVDs.

Marge is sick and

staying home. In how

many ways can Marge

watch the DVDs?

a) State when order

matters.

b) State when order

doesn’t matter.

Bluman, Chapter 4

Permutation Rule

The arrangement of n objects in a specific order

using r objects at a time is called a permutation

of n objects taking r objects at a time. It is

written as nPr and the formula is

n rpn

n r

!

( )!

Counting Rules

Combination is a grouping of objects.

Order does not matter.

Bluman, Chapter 4

!

! !n r

nC

n r r

!

n rP

r

Permutation and combination

Order matters:

nPr

Order does NOT

matter

nCr

Bluman, Chapter 4

Chapter 4


Section 4-4

Example 4-42/4-43

Page #228

23 Bluman, Chapter 4

Example 4-42: Business Locations Suppose a business owner has a choice of 5 locations in

which to establish her business. She decides to rank

each location according to certain criteria, such as price

of the store and parking facilities. How many different

ways can she rank the 5 locations?


first second third fourth fifthchoice choice choice choice choice

120 different ways to rank the locations

5 4 3 2 1

Using factorials, 5! = 120.

Using permutations, 5P5 = 120.

Example 4-43: Business Locations Suppose the business owner in Example 4–42 wishes to

rank only the top 3 of the 5 locations. How many different

ways can she rank them?


first second thirdchoice choice choice

60 different ways to rank the locations

5 4 3

Using permutations, 5P3 = 60.

Chapter 4


Section 4-4

Example 4-44

Page #229


Example 4-44: Television News Stories A television news director wishes to use 3 news stories

on an evening show. One story will be the lead story, one

will be the second story, and the last will be a closing

story. If the director has a total of 8 stories to choose

from, how many possible ways can the program be set

up?

Since there is a lead, second, and closing story, we know

that order matters. We will use permutations.


8 3

8!336

5!P 8 3

3

or 8 7 6 336P

Chapter 4


Section 4-4

Example 4-45

Page #229


Example 4-45: School Musical Plays A school musical director can select 2 musical plays to

present next year. One will be presented in the fall, and

one will be presented in the spring. If she has 9 to pick

from, how many different possibilities are there?

Order matters, so we will use permutations.


9 2

9!72

7!P

9 2

2

or 9 8 72P

Chapter 4


Section 4-4

Example 4-48

Page #231


Example 4-48: School Musicals A newspaper editor has received 8 books to review. He

decides that he can use 3 reviews in his newspaper. How

many different ways can these 3 reviews be selected?

The placement in the newspaper is not mentioned, so

order does not matter. We will use combinations.


8 3

8!8!/ 5!3! 56

5!3!C

8 3

8 7 6or 56

3 2C

8 38 3or 56

3!

PC

Chapter 4


Section 4-4

Example 4-49

Page #231


Example 4-49: Committee Selection In a club there are 7 women and 5 men. A committee of 3

women and 2 men is to be chosen. How many different

possibilities are there?

There are not separate roles listed for each committee

member, so order does not matter. We will use

combinations.

There are 35·10=350 different possibilities.


7 3 5 2

7! 5!Women: 35, Men: 10

4!3! 3!2!C C

On your own:

Read:

TI 83 84 activities on

pages 208 & 235

Summary Table on

page 232

Answer:

Applying Concepts 4-

4 on page 232

Exercises 4-4 starting

with # 1 every other

odd problem.

Bluman, Chapter 4

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4.4 Counting Rules - WordPress.com€¦ · A Question to Ponder: A box contains 3 red chips, 2 blue chips and 5 green chips. A chip is selected, and not replaced and a second chip