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Chapter 11. Section 11.8. Exercise #1. A parent teacher committee consisting of four people is to be formed from 20 parents and five teachers. Find the probability that the committee will consist of the following. (Assume that the selection will be random.). (a) All teachers

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slide2

A parent teacher committee consisting of four people is to be formed from 20 parents and five teachers. Find the probability that the committee will consist of the following. (Assume that the selection will be random.)

slide3

(a) All teachers

(b) Two teachers and two parents

(c) All parents

(d) One teacher and three parents

slide4

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections):

25!

25C4

=

4!(25 – 4)!

slide5

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections):

25!

25C4

=

4!(21!)

slide6

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections):

25 24 23 22 21!

25C4

=

4 3 2 1 21!

slide7

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections):

25C4

= 12,650

slide8

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers):

5!

5C4

=

4!(5 – 4)!

slide9

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers):

5!

5C4

=

4!(1!)

slide10

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers):

5 4!

5C4

=

4! 1

slide11

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers):

5C4

= 5

slide12

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers): 5

n(all teachers)

P(all teachers) =

n(all selections)

slide13

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers): 5

5

P(all teachers) =

12,650

slide14

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers): 5

1

P(all teachers) =

2,530

slide15

A committee of four out of 20 parents and 5 teachers

a) P(all teachers)

n(all selections): 12,650

n(all teachers): 5

P(all teachers) = 0.0004

slide16

n(2 teachers and 2 parents):

5!

20!

5C2

20C2

=

2!(3!)

2!(18!)

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

slide17

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

n(2 teachers and 2 parents):

5 4

20 19

5C2

20C2

=

2 1

2 1

slide18

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

n(2 teachers and 2 parents):

20

380

5C2

20C2

=

2

2

slide19

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

n(2 teachers and 2 parents):

5C2

20C2

= 10 190

slide20

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

n(2 teachers and 2 parents):

5C2

20C2

= 1,900

slide21

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

n(2 teachers and 2 parents): 1,900

P(2 teachers and 2 parents) =

n(2 teachers and 2 parents)

n(all selections)

n(all selections): 12,650

slide22

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

n(2 teachers and 2 parents): 1,900

n(all selections): 12,650

P(2 teachers and 2 parents) =

1,900

12,650

slide23

A committee of four out of 20 parents and 5 teachers

b) P(2 teachers and 2 parents)

P(2 teachers and 2 parents) =0.15

slide24

n(all parents):

20!

20C4

=

4!(16!)

A committee of four out of 20 parents and 5 teachers

c) P(all parents)

slide25

A committee of four out of 20 parents and 5 teachers

c) P(all parents)

n(all parents):

20 19 18 17

20C4

=

4 3 2 1

slide26

A committee of four out of 20 parents and 5 teachers

c) P(all parents)

n(all parents):

116,280

20C4

=

24

slide27

A committee of four out of 20 parents and 5 teachers

c) P(all parents)

n(all parents):

20C4

= 4,845

slide28

A committee of four out of 20 parents and 5 teachers

c) P(all parents)

n(all parents): 4,845

n(all parents)

P(all parents):

n(all selections)

n(all selections): 12,650

slide29

A committee of four out of 20 parents and 5 teachers

c) P(all parents)

n(all parents): 4,845

n(all selections): 12,650

4,845

P(all parents):

12,650

slide30

A committee of four out of 20 parents and 5 teachers

c) P(all parents)

P(all parents): 0.383

slide31

n(1 teacher and 3 parents):

5!

20!

5C1

20C3

=

1!(4!)

3!(17!)

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

slide32

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

n(1 teacher and 3 parents):

5

20 19 18

5C1

20C3

=

1

3 2 1

slide33

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

n(1 teacher and 3 parents):

5

6,840

5C1

20C3

=

1

6

slide34

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

n(1 teacher and 3 parents):

34,200

5C1

20C3

=

6

slide35

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

n(1 teacher and 3 parents):

5C1

20C3

= 5,700

slide36

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

n(1 teacher and 3 parents): 5,700

P(1 teacher and 3 parents) =

n(1 teacher and 3 parents)

n(all selections)

n(all selections): 12,650

slide37

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

n(1 teacher and 3 parents): 5,700

n(all selections): 12,650

P(1 teacher and 3 parents) =

5,700

12,650

slide38

A committee of four out of 20 parents and 5 teachers

d) P(1 teacher and 3 parents)

P(1 teacher and 3 parents) =0.450

slide40

A city council consists of 10 members. Four are Republicans, three are Democrats, and three are Independents. If a committee of three is to be selected, find the probability of selecting

All Republicans

All Democrats

slide41

(c) One of each party

Two Democrats and one Independent

One Independent and two Republicans

slide42

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections):

10!

10C3

=

3!(10 – 3)!

slide43

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections):

10!

10C3

=

3!(7!)

slide44

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections):

10 9 8 7!

10C3

=

3 2 1 7!

slide45

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections):

720

10C3

=

6

slide46

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections):

10C3

= 120

slide47

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R):

4!

4C3

=

3!(4 – 3)!

slide48

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R):

4!

4C3

=

3!(1!)

slide49

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R):

4 3!

4C3

=

3! 1

slide50

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R):

4C3

= 4

slide51

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R): 4

n(all R)

P(all R) =

n(all selections)

slide52

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R): 4

4

P(all R) =

120

slide53

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

n(all selections): 120

n(all R): 4

1

P(all R) =

30

slide54

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

a) P(all R)

P(all R) = 0.033

slide55

n(all D):

3!

3C3

=

3!(3 – 3)!

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

b) P(all D)

slide56

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

b) P(all D)

n(all D):

3!

3C3

=

3!(0)!

slide57

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

b) P(all D)

n(all D):

3C3

= 1

slide58

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

b) P(all D)

n(all D): 1

n(all D)

P(all D) =

n(all selections)

n(all selections): 120

slide59

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

b) P(all D)

n(all D): 1

n(all selections): 120

1

P(all D) =

120

slide60

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

b) P(all D)

P(all D) = 0.0083

slide61

n(one R, one D, one I):

4!

3!

3!

4C1

3C1

3C1

=

1!(3!)

1!(2!)

1!(2!)

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

c) P(one R, one D, one I)

slide62

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

c) P(one R, one D, one I)

n(one R, one D, one I):

4C1

3C1

3C1

= 4 3 3

slide63

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

c) P(one R, one D, one I)

n(one R, one D, one I):

4C1

3C1

3C1

= 36

slide64

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

c) P(one R, one D, one I)

n(one R, one D, one I): 36

P(one R, one D, one I) =

n(one R, one D, one I)

n(all selections)

n(all selections): 120

slide65

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

c) P(one R, one D, one I)

n(one R, one D, one I): 36

n(all selections): 120

P(one R, one D, one I) =

36

120

slide66

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

c) P(one R, one D, one I)

P(one R, one D, one I) =0.300

slide67

n(two D and one I):

3!

3!

3C2

3C1

=

2!(1!)

1!(2!)

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

d) P(two D and one I)

slide68

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

d) P(two D and one I)

n(two D and one I):

3C2

3C1

= 3 3

slide69

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

d) P(two D and one I)

n(two D and one I):

3C2

3C1

= 9

slide70

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

d) P(two D and one I)

n(two D and one I): 9

n(two D and one I)

P(two D and one I) =

n(all selections)

n(all selections): 120

slide71

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

d) P(two D and one I)

n(two D and one I): 9

n(all selections): 120

9

P(two D and one I) =

120

slide72

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

d) P(two D and one I)

P(two D and one I) = 0.075

slide73

n(one I and two R):

3!

4!

3C1

4C2

=

1!(2!)

2!(2!)

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

slide74

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

n(one I and two R):

4 3

3C1

4C2

= 3

2 2

slide75

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

n(one I and two R):

12

3C1

4C2

= 3

2

slide76

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

n(one I and two R):

3C1

4C2

= 3 6

slide77

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

n(one I and two R):

3C1

4C2

= 18

slide78

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

n(one I and two R): 18

n(one I and two R)

P(one I and two R) =

n(all selections)

n(all selections): 120

slide79

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

n(one I and two R): 18

n(all selections): 120

18

P(one I and two R) =

120

slide80

A committee of 3 out of 10 members: 4 Republicans (R), 3 Democrats (D), 3 Independents (I)

e) P(one I and two R)

P(one I and two R) =0.15

slide82

Find the probability of selecting three science books and four math books from eight science books and nine math books. The books are selected at random.

slide83

n(3 S and 4 M):

8!

9!

8C3

9C4

=

3!(5!)

4!(5!)

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

slide84

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M):

8 7 6 5!

9 8 7 6 5!

8C3

9C4

=

3 2 1 5!

4 3 2 1 5!

slide85

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M):

336

3024

8C3

9C4

=

6

24

slide86

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M):

8C3

9C4

= 56 126

slide87

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M):

8C3

9C4

= 7056

slide88

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M): 7,056

n(all selections):

17!

17C7

=

7!(10!)

slide89

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M): 7,056

n(all selections):

17 16 15 14 13 12 11 10!

17C7

=

7 6 5 4 3 2 1 10!

slide90

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M): 7,056

n(all selections):

98,017,920

17C7

=

5040

slide91

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M): 7,056

n(all selections):

17C7

= 19,448

slide92

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M): 7,056

n(3 S and 4 M)

P(3 S and 4 M) =

n(all selections)

n(all selections): 19,448

slide93

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

n(3 S and 4 M): 7,056

n(all selections): 19,448

7,056

P(3 S and 4 M) =

19,448

slide94

Select 3 science books (S) and 4 math books (M) from 8 science books and 9 math books.

P(3 S and 4 M) =0.363

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