1. Arches National Park is located in southern Utah. The park is famous for its beautiful desert landscape and it many natural sandstone arches. Park Ranger Edward McCarrick started an inventory of natural arches within the park that have an opening of at least 3 feet. The following table is based on information taken from the book Canyon County Arches and Bridges, by F.A. Barnes. The height of the arch opening is rounded to the nearest foot.
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Height of arch, feet
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3-9
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10-29
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30-49
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50-74
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75 and higher
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|
Number of arches in park
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111
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96
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30
|
33
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18
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For an arch chosen at random, use the preceding information to estimate the probability that the height of the opening arch is
(a) 3 to 9 feet tall
(b) 30 feet or taller
(c) 3 to 49 feet tall
(d) 10 to 74 feet tall
(e) 75 feet or taller
2. Hospital records indicated that maternity patients stayed in the hospital for the
number of days shown in the distribution.
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Number of days stayed
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Frequency
|
|
3
|
15
|
|
4
|
32
|
|
5
|
56
|
|
6
|
19
|
|
7
|
5
|
Find these probabilities:
a) A patient stayed exactly 5 days.
b) A patient stayed less than 6 days
c) A patient stayed at most 4 days.
d) A patient stayed at least 5 days.
3. A large department store has 500 employees. There are 350 females and 200 of them are under the age of 25. There are 75 males under 25. If an employee is selected for promotion, find the probability that the employee will be one of the following:
a. Under 25 and female
b. Over 24 or a female
c. Male or over 24.
d. Over 24, given the employee is a female.
4. An urn contains five balls, three of which are red and two of which are blue. You choose two balls at random. You replace the first ball before drawing the second ball.
What is the probability that the first ball is red?
What is the probability that the first ball is red and the second ball is blue?
What is the probability of choosing at least one blue?
5. A family owns two cars. The probabilities that cars S and T will fail to start on a cold morning are 2/10 and 3/10, respectively. Assuming that the failure of one car to start is independent of the starting of the second car, find the probabilty that on a cold morning:
(a) both cars will fail to start
(b) at least one of the cars will fail to start
(c) exactly one of the cars will fail to start
6. Suppose the probability that a particular type of smoke detector will function properly and sound an alarm in the presence of smoke is 0.7. If you have two such alarms in your home, what is the probability that at least one functions when a fire occurs? Assume that the smoke detectors operate independent of one another.
7. The following table describes the adult population of a small suburb of a large southern city.
Income
......................................................................
Under $20,000 $20,000-$50,000 Over $50,000
__________________________________________________________
Under 25 950 1,000 50
Age 25-45 450 2,050 1,500
Over 45 50 950 1,000
__________________________________________________________
A marketing research firm plans to randomly select one adult from this suburb to evaluate a new food product. Consider the following events:
A: Person is under 25
B: Person is between 25 and 45
C: Person is over 45
D: Person has income under $20,000
E: Person has income of $20,000-$50,000
F: Person has income over $50,000
Find the following probabilities.
a) P(B)
b) P(F)
c) P(C and F)
d) P(B or C)
e) P(not A)
f) P(F | C)
g) P(B | E)
h) List the pairs of events that are mutually exclusive.
- Two special dice are made. One has eight sides numbered 1 through 8, and the other has ten sides, numbered 1 through 10. These two dice are tossed. What is the probability of rolling a sum of 13?
9. The Mass Millions lottery game involved selecting six numbers (no repetitions and order does not matter) from a collection of 46 numbers, {1,2,3,...,44,45,46}.
Let the sample space S consist of all possible six-number combinations that can be selected. Suppose you select six numbers. The number of elements in S that match 0, 1, 2, 3, 4, 5, or 6 numbers of the winning combinations are
Number of Matches Number of Possibilities
0 3,838,380
1 3,948,048
2 1,370,850
3 197,600
4 11,700
5 240
6 1
What is the probability of matching at least three numbers?
10. A hospital administrator records a 0 if a patient has no medical insurance and 1 if
the patient does have medical insurance. The administrator also records an A, B, C,
D, or E, representing good, fair, poor, serious, or critical condition, respectively.
List the outcomes of the sample space S.
List the outcomes of the event consisting of a selected patient
(a) with no medical insurance and in serious or critical condition
(b) with medical insurance and not in critical condition
(c) in good or fair condition
(d) with medical insurance
Find the probability of each event in (a)-(d)
11. A study is to be made in a large university to try to determine a relationship, if any,
between the gender of a faculty member and his or her salary. Faculty are to be interviewed and classified according to gender and salary category. Suppose that M=Male, F=Female, 1=less than $30,000, 2=less than $35,000 but greater than or equal to $30,000, 3= less than $40,000 but greater than or equal to $35,000, 4=less than $45,000 but greater than or equal to $40,000, 5=less than $50,000 but greater than or equal to $45,000, and 6=greater than or equal to $50,000.
List the outcomes of the sample space S.
List the outcomes of the event consisting of a selected faculty member
(a) with salary less than $40,000
(b) who is female or has a salary greater than or equal to $40,000
(c) who is a male with a salary greater than or equal to $50,000
(d) who is a male with a salary less than $40,000 and greater than or equal to $35,000
12. Consider the tossing of two fair dice. Let A and B be the following events:
A: sum is 7 or more
B: sum is less than 11
Find P(A), P(B), P(A and B), and P(A or B).
13. At a meeting of a college student government council, 50 students are present: 20
freshmen, 15 sophomores, 10 juniors, and 5 seniors. One student is randomly selected to deliver a petition to the school administration. Call A the event that occurs of the student is a freshmen and B the event that the student is a sophomore.
What outcomes are in the sample space?
What outcomes are in event A and event B?
Assume all outcomes are equally likely. What is P(A) and P(B)?
What is the probability of selecting a freshmen or a sophomore?