1. Chapter 4
Probability and Counting Rules
Section 4-2
Sample Spaces and Probability

2. A sum less than or equal to 4
Section 4-2 Exercise #13
If two dice are rolled one time, find the probability of getting these results.
A sum of 6
Doubles
A sum of 7 or 11
A sum greater than 9
A sum less than or equal to 4

3. A sum of 6
Total of 36 outcomes
Doubles
There are six ways to get doubles. They are (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6).

4. Total of 36 outcomes
A sum of 7 or 11
There are six ways to get a sum of 7. They are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
There are two ways to get a sum of 11. They are (5,6) and (6,5).
A sum of greater than 9
To get a sum greater than nine, one must roll a 10, 11, or 12.
There are six ways to get a 10, 11, or 12. They are (4,6), (5,5), (6,4), (6,5), (5,6), and (6,6).

6. Number of Tests Performed
e. The patient has had 1 or 2 tests done.
Number of Tests Performed
Number of Patients 12 1 8 2 3
4 or more 5

7. Chapter 4
Probability and Counting Rules
Section 4-3
Exercise #23

8. If one card is drawn from an ordinary deck of cards, find the probability of getting the following:
A king or a queen or a jack.
A club or a heart or a spade.
A king or a queen or a diamond.
An ace or a diamond or a heart.
A 9 or a 10 or a spade or a club.

9. If one card is drawn from an ordinary deck of cards, find the probability of getting the following:
A king or a queen or a jack.
There are 4 kings, 4 queens, and 4 jacks,
hence:
P (king or queen or jack)

10. If one card is drawn from an ordinary deck of cards, find the probability of getting the following:
A club or a heart or a spade.
There are 13 clubs, 13 hearts, and 13
spades, hence:
P(club or heart or spade)

11. If one card is drawn from an ordinary deck of cards, find the probability of getting the following:
A king or a queen or a diamond.
There are 4 kings, 4 queens, and 13
diamonds but the king and queen of
diamonds were counted twice, hence:
P(king or queen or diamond)
P(king) + P(queen) + P(diamond)
– P(king or queen of diamonds)

12. An ace or a diamond or a heart. There are 4 aces, 13 diamonds and 13
If one card is drawn from an ordinary deck of cards, find the probability of getting the following:
An ace or a diamond or a heart.
There are 4 aces, 13 diamonds and 13
hearts. There is one ace of diamonds
and one ace of hearts, hence:
P(ace or diamond or heart)

13. If one card is drawn from an ordinary deck of cards, find the probability of getting the following:
A 9 or a 10 or a spade or a club.
There are 4 nines, 4 tens, 13 spades, and 13 clubs. There is one nine of spades, one ten of spades, one nine of clubs, and one ten of clubs, hence:
P ( 9 or 10 or spade or club)

14. If one card is drawn from an ordinary deck of cards, find the probability of getting the following:
A 9 or a 10 or a spade or a club.
P ( 9 or 10 or spade or club)

15. Chapter 4
Probability and Counting Rules
Section 4-4
The Multiplication Rules and Conditional Probability

16. Chapter 4
Probability and Counting Rules
Section 4-4
Exercise #7

17. At a local university 54.3% of incoming first-year students have computers. If three students are selected at random, find the following probabilities.
None have computers
At least one has a computer
All have computers

18. At a local university 54.3% of incoming first-year students have computers. If three students are selected at random, find the following probabilities.
None have computers

19. At a local university 54.3% of incoming first-year students have computers. If three students are selected at random, find the following probabilities.
b. At least one has a computer

20. At a local university 54.3% of incoming first-year students have computers. If three students are selected at random, find the following probabilities.
c. All have computers

21. Chapter 4
Probability and Counting Rules
Section 4-4
Exercise #21

22. D (0.8)(0.1) = 0.08
0.8 ND
D (0.2)(0.18) = 0.036
0.2 ND

23. Finally, use the addition rule, since the item is
chosen at random from model I or model II.
D (0.8)(0.1) = 0.08
0.8 ND
D (0.2)(0.18) = 0.036
0.2 ND

24. Chapter 4
Probability and Counting Rules
Section 4-4
Exercise #31

25. In Rolling Acres Housing Plan, 42% of the houses have
a deck and a garage; 60% have a deck. Find the
probability that a home has a garage, given
that it has a deck.

26. Chapter 4
Probability and Counting Rules
Section 4-4
Exercise #35

27. Consider this table concerning utility
Corporation
Government
Individual
U.S.
70,894
921
6129
Foreign
63,182
104
6267
Consider this table concerning utility
patents granted for a specific year.
Select one patent at random.
What is the probability that it is a
foreign patent, given that it was
issued to a corporation?
What is the probability that it
was issued to an individual,
given that it was a U.S. patent?

28. What is the probability that it is a foreign patent, given that it was
Corporation
Government
Individual
U.S.
70,894
921
6129
Foreign
63,182
104
6267
What is the probability that it is a
foreign patent, given that it was
issued to a corporation?
P(foreign patent | corporation)

29. P(foreign patent | corporation)
Government
Individual
U.S.
70,894
921
6129
Foreign
63,182
104
6267
P(foreign patent | corporation)

30. P(foreign patent | corporation)
Government
Individual
U.S.
70,894
921
6129
Foreign
63,182
104
6267
P(foreign patent | corporation)

31. What is the probability that it was
Corporation
Government
Individual
U.S.
70,894
921
6129
Foreign
63,182
104
6267
What is the probability that it was
issued to an individual, given that
it was a U.S. patent?
P (individual | U.S.)

32. P(individual | U.S.) Corporation Government Individual U.S. 70,894 921
6129
Foreign
63,182
104
6267
P(individual | U.S.)

33. P(individual | U.S.) Corporation Government Individual U.S. 70,894 921
6129
Foreign
63,182
104
6267
P(individual | U.S.)

34. Chapter 4
Probability and Counting Rules
Section 4-5
Counting Rules

35. Chapter 4
Probability and Counting Rules
Section 4-5
Exercise #9

36. How many different 3 - digit identification tags can be made if the digits can be used more than once? If the first digit must be a 5 and repetitions are not permitted?
If digits can be used more than once: Since there are three spaces to fill and 10 choices for each space, the solution is:

37. How many different 3 - digit identification tags can be made if the digits can be used more than once? If the first digit must be a 5 and repetitions are not permitted?
If the first digit must be a 5 and repetitions are not permitted: There is only one way to assign the first digit, 9 ways to assign the second, and 8 ways to assign the third:

38. Chapter 4
Probability and Counting Rules
Section 4-5
Exercise #21

39. How many different ID cards can be made if there are 6 digits on a card and no digit can be used more than once?
Since order is important, the solution is:

40. Chapter 4
Probability and Counting Rules
Section 4-5
Exercise #31

41. How many ways can a committee of 4 people be selected from a group of 10 people?
Since order is not important, the solution is:

42. Chapter 4
Probability and Counting Rules
Section 4-5
Exercise #41

43. How many ways can a foursome of 2 men and 2 women
be selected from 10 men and 12 women in a golf club?

44. Chapter 4
Probability and Counting Rules
Section 4-6
Probability and Counting Rules

45. Chapter 4
Probability and Counting Rules
Section 4-6
Exercise #3

46. In a company there are 7 executives: 4 women and 3 men
In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities.
All 3 selected will be women.
All 3 selected will be men.
c. 2 men and 1 woman will be selected.
d. 1 man and 2 women will be selected.

47. In a company there are 7 executives: 4 women and 3 men
In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities.
All 3 selected will be women.

48. In a company there are 7 executives: 4 women and 3 men
In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities.
All 3 selected will be men.

49. In a company there are 7 executives: 4 women and 3 men
In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities.
c. 2 men and 1 woman will be selected.

50. In a company there are 7 executives: 4 women and 3 men
In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities.
d. 1 man and 2 women will be selected.

51. Chapter 4
Probability and Counting Rules
Section 4-6
Exercise #9

52. A committee of 4 people is to be formed from 6 doctors and 8 dentists
A committee of 4 people is to be formed from 6 doctors and 8 dentists. Find the probability that the committee will consist of:
All dentists.
b. 2 dentists and 2 doctors.
c. All doctors.
d. 3 doctors and 1 dentist.
e. 1 doctor and 3 dentists.

53. A committee of 4 people is to be formed from 6 doctors and 8 dentists
A committee of 4 people is to be formed from 6 doctors and 8 dentists. Find the probability that the committee will consist of:
All dentists.

54. A committee of 4 people is to be formed from 6 doctors and 8 dentists
A committee of 4 people is to be formed from 6 doctors and 8 dentists. Find the probability that the committee will consist of:
b. 2 dentists and 2 doctors.

55. A committee of 4 people is to be formed from 6 doctors and 8 dentists
A committee of 4 people is to be formed from 6 doctors and 8 dentists. Find the probability that the committee will consist of:
c. All doctors.

56. A committee of 4 people is to be formed from 6 doctors and 8 dentists
A committee of 4 people is to be formed from 6 doctors and 8 dentists. Find the probability that the committee will consist of:
d. 3 doctors and 1 dentist.

57. A committee of 4 people is to be formed from 6 doctors and 8 dentists
A committee of 4 people is to be formed from 6 doctors and 8 dentists. Find the probability that the committee will consist of:
e. 1 doctor and 3 dentists.

58. Chapter 4
Probability and Counting Rules
Section 4-6
Exercise #11

59. A drawer contains 11 identical red socks and 8 identical black socks
A drawer contains 11 identical red socks and 8 identical black socks. Suppose that you choose 2 socks at random in the dark.
What is the probability that you get a pair of red socks?
What is the probability that you get a pair of black socks?
What is the probability that you get 2 unmatched socks?
Where did the other red sock go?

60. A drawer contains 11 identical red socks and 8 identical black socks
A drawer contains 11 identical red socks and 8 identical black socks. Suppose that you choose 2 socks at random in the dark.
What is the probability that you get a pair of red socks?

61. A drawer contains 11 identical red socks and 8 identical black socks
A drawer contains 11 identical red socks and 8 identical black socks. Suppose that you choose 2 socks at random in the dark.
b. What is the probability that you get a pair of black socks?

62. A drawer contains 11 identical red socks and 8 identical black socks
A drawer contains 11 identical red socks and 8 identical black socks. Suppose that you choose 2 socks at random in the dark.
c. What is the probability that you get 2 unmatched socks?

63. A drawer contains 11 identical red socks and 8 identical black socks
A drawer contains 11 identical red socks and 8 identical black socks. Suppose that you choose 2 socks at random in the dark.
d. Where did the other red sock go?
It probably got lost in the wash!

64. Chapter 4
Probability and Counting Rules
Section 4-6
Exercise #15

65. Find the probability that if 5 different- sized washers are arranged in a row, they will be arranged in order of size.
There are 5! = 120 ways to arrange 5 washers in a row and 2 ways to have them in correct order, small to large or large to small; hence, the probability is: