PROBLEM
college plans to interview 8 students for possible offer of graduate assistantships
A college plans
to interview 8 students for possible offer of graduate assistantships. The college has three assistantships available. How many groups of three can the college select?
- A student has to take 9 more courses before he can graduate. If none of the courses are prerequisite to others, how many groups of four courses can he select for the next semester?
- From among 8 students how many committees consisting of 3 students can be selected?
- From a group of seven finalists to a contest, three individuals are to be selected for the first and second and third places. Determine the number of possible selections.
- Ten individuals are candidates for positions of president, vice president of an organization. How many possibilities of selections exist?
- Assume you have applied for two jobs A and B. The probability that you get an offer for job A is 0.23. The probability of being offered job B is 0.19. The probability of getting at least one of the jobs is 0.38.
a.
What is the probability that you will be offered both jobs?
b.
Are events A and B mutually exclusive? Why or why not? Explain.
- Assume you have applied for two scholarships, a Merit scholarship (M) and an Athletic scholarship (A). The probability that you receive an Athletic scholarship is 0.18. The probability of receiving both scholarships is 0.11. The probability of getting at least one of the scholarships is 0.3.
a.
What is the probability that you will receive a Merit scholarship?
b.
Are events A and M mutually exclusive? Why or why not? Explain.
c.
Are the two events A, and M, independent? Explain, using probabilities.
d.
What is the probability of receiving the Athletic scholarship given that you have been awarded the Merit scholarship?
e.
What is the probability of receiving the Merit scholarship given that you have been awarded the Athletic scholarship?
- A survey of a sample of business students resulted in the following information regarding the genders of the individuals and their selected major.
Selected Major
Gender
Management
Marketing
Others
Total
Male
40
10
30
80
Female
30
20
70
120
Total
70
30
100
200
a.
What is the probability of selecting an individual who is majoring in Marketing?
b.
What is the probability of selecting an individual who is majoring in Management, given that the person is female?
c.
Given that a person is male, what is the probability that he is majoring in Management?
d.
What is the probability of selecting a male individual?
- Sixty percent of the student body at UTC is from the state of Tennessee (T), 30% percent are from other states (O), and the remainder are international students (I). Twenty percent of students from Tennessee live in the dormitories, whereas, 50% of students from other states live in the dormitories. Finally, 80% of the international students live in the dormitories.
a.
What percentage of UTC students live in the dormitories?
b.
Given that a student lives in the dormitory, what is the probability that she/he is an international student?
c.
Given that a student lives in the dormitory, what is the probability that she/he is from Tennessee?
- The probability of an economic decline in the year 2008 is 0.23. There is a probability of 0.64 that we will elect a republican president in the year 2008. If we elect a republican president, there is a 0.35 probability of an economic decline. Let "D" represent the event of an economic decline, and "R" represent the event of election of a Republican president.
a.
Are "R" and "D" independent events?
b.
What is the probability of a Republican president and economic decline in the year 2008?
c.
If we experience an economic decline in the year 2008, what is the probability that there will a Republican president?
d.
What is the probability of economic decline or a Republican president in the year 2008? Hint: You want to find P(D È R).
- As a company manager for Claimstat Corporation there is a 0.40 probability that you will be promoted this year. There is a 0.72 probability that you will get a promotion, a raise, or both. The probability of getting a promotion and a raise is 0.25.
a.
If you get a promotion, what is the probability that you will also get a raise?
b.
What is the probability that you will get a raise?
c.
Are getting a raise and being promoted independent events? Explain using probabilities.
d.
Are these two events mutually exclusive? Explain using probabilities.
- A company plans to interview 10 recent graduates for possible employment. The company has three positions open. How many groups of three can the company select?
- A student has to take 7 more courses before she can graduate. If none of the courses are prerequisites to others, how many groups of three courses can she select for the next semester?
- How many committees, consisting of 3 female and 5 male students, can be selected from a group of 5 female and 8 male students?
- Six vitamin and three sugar tablets identical in appearance are in a box. One tablet is taken at random and given to Person A. A tablet is then selected and given to Person B. What is the probability that
a.
Person A was given a vitamin tablet?
b.
Person B was given a sugar tablet given that Person A was given a vitamin tablet?
c.
neither was given vitamin tablets?
d.
both were given vitamin tablets?
e.
exactly one person was given a vitamin tablet?
f.
Person A was given a sugar tablet and Person B was given a vitamin tablet?
g.
Person A was given a vitamin tablet and Person B was given a sugar tablet?
- The sales records of a real estate agency show the following sales over the past 200 days:
Number of
Number
Houses Sold
of Days
0
60
1
80
2
40
3
16
4
4
a.
How many sample points are there?
b.
Assign probabilities to the sample points and show their values.
c.
What is the probability that the agency will not sell any houses in a given day?
d.
What is the probability of selling at least 2 houses?
e.
What is the probability of selling 1 or 2 houses?
f.
What is the probability of selling less than 3 houses?
- A bank has the following data on the gender and marital status of 200 customers.
Male
Female
Single
20
30
Married
100
50
a.
What is the probability of finding a single female customer?
b.
What is the probability of finding a married male customer?
c.
If a customer is female, what is the probability that she is single?
d.
What percentage of customers is male?
e.
If a customer is male, what is the probability that he is married?
f.
Are gender and marital status mutually exclusive?
g.
Is marital status independent of gender? Explain using probabilities.
- An applicant has applied for positions at Company A and Company B. The probability of getting an offer from Company A is 0.4, and the probability of getting an offer from Company B is 0.3. Assuming that the two job offers are independent of each other, what is the probability that
a.
the applicant gets an offer from both companies?
b.
the applicant will get at least one offer?
c.
the applicant will not be given an offer from either company?
d.
Company A does not offer her a job, but Company B does?
- An experiment consists of throwing two six-sided dice and observing the number of spots on the upper faces. Determine the probability that
a.
the sum of the spots is 3.
b.
each die shows four or more spots.
c.
the sum of the spots is not 3.
d.
neither a one nor a six appear on each die.
e.
a pair of sixes appear.
f.
the sum of the spots is 7.
- Two of the cylinders in an eight-cylinder car are defective and need to be replaced. If two cylinders are selected at random, what is the probability that
a.
both defective cylinders are selected?
b.
no defective cylinder is selected?
c.
at least one defective cylinder is selected?
- Assume two events A and B are mutually exclusive and, furthermore, P(A) = 0.2 and P(B) = 0.4.
a.
Find P(A Ç B).
b.
Find P(A È B).
c.
Find P(A ½ B).
- A government agency has 6,000 employees. The employees were asked whether they preferred a four-day work week (10 hours per day), a five-day work week (8 hours per day), or flexible hours. You are given information on the employees' responses broken down by sex.
Male
Female
Total
Four days
300
600
900
Five days
1,200
1,500
2,700
Flexible
300
2,100
2,400
Total
1,800
4,200
6,000
a.
What is the probability that a randomly selected employee is a man and is in favor of a four-day work week?
b.
What is the probability that a randomly selected employee is female?
c.
A randomly selected employee turns out to be female. Compute the probability that she is in favor of flexible hours.
d.
What percentage of employees is in favor of a five-day work week?
e.
Given that a person is in favor of flexible time, what is the probability that the person is female?
f.
What percentage of employees is male and in favor of a five-day work week?
- Forty percent of the students who enroll in a statistics course go to the statistics laboratory on a regular basis. Past data indicates that 65% of those students who use the lab on a regular basis make a grade of A in the course. On the other hand, only 10% of students who do not go to the lab on a regular basis make a grade of A. If a particular student made an A, determine the probability that she or he used the lab on a regular basis.
- A corporation has 15,000 employees. Sixty-two percent of the employees are male. Twenty-three percent of the employees earn more than $30,000 a year. Eighteen percent of the employees are male and earn more than $30,000 a year.
a.
If an employee is taken at random, what is the probability that the employee is male?
b.
If an employee is taken at random, what is the probability that the employee earns more than $30,000 a year?
c.
If an employee is taken at random, what is the probability that the employee is male and earns more than $30,000 a year?
d.
If an employee is taken at random, what is the probability that the employee is male or earns more than $30,000 a year?
e.
The employee taken at random turns out to be male. Compute the probability that he earns more than $30,000 a year.
f.
Are being male and earning more than $30,000 a year independent?
- In the two upcoming basketball games, the probability that UTC will defeat Marshall is 0.63, and the probability that UTC will defeat Furman is 0.55. The probability that UTC will defeat both opponents is 0.3465.
a.
What is the probability that UTC will defeat Furman given that they defeat Marshall?
b.
What is the probability that UTC will win at least one of the games?
c.
What is the probability of UTC winning both games?
d.
Are the outcomes of the games independent? Explain and substantiate your answer.
- A small town has 5,600 residents. The residents in the town were asked whether or not they favored building a new bridge across the river. You are given the following information on the residents' responses, broken down by sex.
Men
Women
Total
In Favor
1,400
280
1,680
Opposed
840
3,080
3,920
Total
2,240
3,360
5,600
Let:
M be the event a resident is a man
W be the event a resident is a woman
F be the event a resident is in favor
P be the event a resident is opposed
a.
Find the joint probability table.
b.
Find the marginal probabilities.
c.
What is the probability that a randomly selected resident is a man and is in favor of building the bridge?
d.
What is the probability that a randomly selected resident is a man?
e.
What is the probability that a randomly selected resident is in favor of building the bridge?
f.
What is the probability that a randomly selected resident is a man or in favor of building the bridge?
g.
A randomly selected resident turns out to be male. Compute the probability that he is in favor of building the bridge.
- On a recent holiday evening, a sample of 500 drivers was stopped by the police. Three hundred were under 30 years of age. A total of 250 were under the influence of alcohol. Of the drivers under 30 years of age, 200 were under the influence of alcohol.
Let A be the event that a driver is under the influence of alcohol.
Let Y be the event that a driver is less than 30 years old.
a.
Determine P(A) and P(Y).
b.
What is the probability that a driver is under 30 and not under the influence of alcohol?
c.
Given that a driver is not under 30, what is the probability that he/she is under the influence of alcohol?
d.
What is the probability that a driver is under the influence of alcohol, when we know the driver is under 30?
e.
Show the joint probability table.
f.
Are A and Y mutually exclusive events? Explain.
g.
Are A and Y independent events? Explain.
- You are given the following information on Events A, B, C, and D.
P(A) = .4
P(A È D) = .6
P(B) = .2
P(A½B) = .3
P(C) = .1
P(A Ç C) = .04
P(A Ç D) = .03
a.
Compute P(D).
b.
Compute P(A Ç B).
c.
Compute P(A½C).
d.
Compute the probability of the complement of C.
e.
Are A and B mutually exclusive? Explain your answer.
f.
Are A and B independent? Explain your answer.
g.
Are A and C mutually exclusive? Explain your answer.
h.
Are A and C independent? Explain your answer.
- In a city, 60% of the residents live in houses and 40% of the residents live in apartments. Of the people who live in houses, 20% own their own business. Of the people who live in apartments, 10% own their own business. If a person owns his or her own business, find the probability that he or she lives in a house.
- Four workers at a fast food restaurant pack the take-out chicken dinners. John packs 45% of the dinners but fails to include a salt packet 4% of the time. Mary packs 25% of the dinners but omits the salt 2% of the time. Sue packs 30% of the dinners but fails to include the salt 3% of the time. You have purchased a dinner and there is no salt.
a.
Find the probability that John packed your dinner.
b.
Find the probability that Mary packed your dinner.
- A statistics professor has noted from past experience that a student who follows a program of studying two hours for each hour in class has a probability of 0.9 of getting a grade of C or better, while a student who does not follow a regular study program has a probability of 0.2 of getting a C or better. It is known that 70% of the students follow the study program. Find the probability that a student who has earned a C or better grade, followed the program.
- All the employees of ABC Company are assigned ID numbers. The ID number consists of the first letter of an employee's last name, followed by four numbers.
a.
How many possible different ID numbers are there?
b.
How many possible different ID numbers are there for employees whose last name starts with an A?
- Assume you have applied to two different universities (let's refer to them as Universities A and B) for your graduate work. In the past, 25% of students (with similar credentials as yours) who applied to University A were accepted, while University B accepted 35% of the applicants. Assume events are independent of each other.
a.
What is the probability that you will be accepted in both universities?
b.
What is the probability that you will be accepted to at least one graduate program?
c.
What is the probability that one and only one of the universities will accept you?
d.
What is the probability that neither university will accept you?
- The following table shows the number of students in three different degree programs and whether they are graduate or undergraduate students:
Undergraduate
Graduate
Total
Business
150
50
200
Engineering
150
25
175
Arts & Sciences
100
25
125
Total
400
100
500
a.
What is the probability that a randomly selected student is an undergraduate?
b.
What percentage of students is engineering majors?
c.
If we know that a selected student is an undergraduate, what is the probability that he or she is a business major?
d.
A student is enrolled in the Arts and Sciences school. What is the probability that the student is an undergraduate student?
e.
What is the probability that a randomly selected student is a graduate Business major?
- A survey of business students who had taken the Graduate Management Admission Test (GMAT) indicated that students who have spent at least five hours studying GMAT review guides have a probability of 0.85 of scoring above 400. Students who do not review have a probability of 0.65 of scoring above 400. It has been determined that 70% of the business students review for the test.
a.
Find the probability of scoring above 400.
b.
Find the probability that a student who scored above 400 reviewed for the test.
- A machine is used in a production process. From past data, it is known that 97% of the time the machine is set up correctly. Furthermore, it is known that if the machine is set up correctly, it produces 95% acceptable (non-defective) items. However, when it is set up incorrectly, it produces only 40% acceptable items.
a.
An item from the production line is selected. What is the probability that the selected item is non-defective?
b.
Given that the selected item is non-defective, what is the probability that the machine is set up correctly?
- A committee of 4 is to be selected from a group of 12 people. How many possible committees can be selected?
- Assume a businessman has 7 suits and 8 ties. He is planning to take 3 suits and 2 ties with him on his next business trip. How many possibilities of selection does he have?
- The results of a survey of 800 married couples and the number of children they had is shown below.
Number of Children
Probability
0
0.050
1
0.125
2
0.600
3
0.150
4
0.050
5
0.025
If a couple is selected at random, what is the probability that the couple will have
a.
Less than 4 children?
b.
More than 2 children?
c.
Either 2 or 3 children?
- Assume that in your hand you hold an ordinary six-sided die and a dime. You toss both the die and the dime on a table.
a.
What is the probability that a head appears on the dime and a six on the die?
b.
What is the probability that a tail appears on the dime and any number more than 3 on the die?
c.
What is the probability that a number larger than 2 appears on the die?
- A very short quiz has one multiple choice question with five possible choices (a, b, c, d, e) and one true or false question. Assume you are taking the quiz but do not have any idea what the correct answer is to either question, but you mark an answer anyway.
a.
What is the probability that you have given the correct answer to both questions?
b.
What is the probability that only one of the two answers is correct?
c.
What is the probability that neither answer is correct?
d.
What is the probability that only your answer to the multiple choice question is correct?
e.
What is the probability that you have only answered the true or false question correctly?
- Assume that each year the IRS randomly audits 10% of the tax returns. If a married couple has filed separate returns,
a.
What is the probability that both the husband and the wife will be audited?
b.
What is the probability that only one of them will be audited?
c.
What is the probability that neither one of them will be audited?
d.
What is the probability that at least one of them will be audited?
- Tammy is a general contractor and has submitted two bids for two projects (A and B). The probability of getting project A is 0.65. The probability of getting project B is 0.77. The probability of getting at least one of the projects is 0.90.
a.
What is the probability that she will get both projects?
b.
Are the events of getting the two projects mutually exclusive? Explain, using probabilities.
c.
Are the two events independent? Explain, using probabilities.
- Assume you are taking two courses this semester (A and B). The probability that you will pass course A is 0.835, the probability that you will pass both courses is 0.276. The probability that you will pass at least one of the courses is 0.981.
a.
What is the probability that you will pass course B?
b.
Is the passing of the two courses independent events? Use probability information to justify your answer.
c.
Are the events of passing the courses mutually exclusive? Explain.
- In a random sample of UTC students 50% indicated they are business majors, 40% engineering majors, and 10% other majors. Of the business majors, 60% were females; whereas, 30% of engineering majors were females. Finally, 20% of the other majors were female.
a.
What percentage of students in this sample was female?
b.
Given that a person is female, what is the probability that she is an engineering major?
- In a recent survey in a Statistics class, it was determined that only 60% of the students attend class on Fridays. From past data it was noted that 98% of those who went to class on Fridays pass the course, while only 20% of those who did not go to class on Fridays passed the course.
a.
What percentage of students is expected to pass the course?
b.
Given that a person passes the course, what is the probability that he/she attended classes on Fridays?
- You are applying for graduate school at University A. In the past 42% of the applicants to this university have been accepted. It is also known that 70% of those students who have been accepted have had GMAT scores in excess of 550 while 40% of the students who were not accepted had GMAT scores in excess of 550. You take the GMAT exam and score 640. What is the probability that you will be accepted into graduate school of university A?
:
- In a recent survey about appliance ownership, 58.3% of the respondents indicated that they own Maytag appliances, while 23.9% indicated they own both Maytag and GE appliances and 70.7% said they own at least one of the two appliances.
Define the events as
M = Owning a Maytag appliance
G = Owning a GE appliance
a.
What is the probability that a respondent owns a GE appliance?
b.
Given that a respondent owns a Maytag appliance, what is the probability that the respondent also owns a GE appliance?
c.
Are events "M" and "G" mutually exclusive? Why or why not?
Explain, using probabilities.
d.
Are the two events "M" and "G" independent?
Explain, using probabilities.
- Records of a company show that 20% of the employees have only a high school diploma; 70% have bachelor degrees; and 10% have graduate degrees. Of those with only a high school diploma, 10% hold management positions; whereas, of those having bachelor degrees, 40% hold management positions. Finally, 80% of the employees who have graduate degrees hold management positions.
a.
What percentage of employees holds management positions?
b.
Given that a person holds a management position, what is the probability that she/he has a graduate degree?
- From a group of three finalists for a privately endowed scholarship, two individuals are to be selected for the first and second places. Determine the number of possible selections.
- Eight individuals are candidates for positions of president, vice president, and treasurer of an organization. How many possibilities of selections exist?
- In a horse race, nine horses are running. Assume you have purchased a Trifecta ticket. (In Trifecta, the player selects three horses as first, second, and third place winners. To win, those three horses must finish the race in the precise order the player has selected.) How many possibilities of a Trifecta exist?
- An automobile dealer has kept records on the customers who visited his showroom. Forty percent of the people who visited his dealership were female. Furthermore, his records show that 35% of the females who visited his dealership purchased an automobile, while 20% of the males who visited his dealership purchased an automobile. Let
A1 = the event that the customer is female
A2 = the event that the customer is male
a.
What is the probability that a customer entering the showroom will buy an automobile?
b.
A car salesperson has just informed us that he sold a car to a customer. What is the probability that the customer was female?
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