Exam 2 Review Math 141

 

1. Set up and solve the following linear programming problem. Find any leftover resources.

A farmer has at most 40 acres on which to plant two crops A and B. He wants the number of acres of crop B to be at least 25% of the total acres planted. His cultivation costs are $30 per acre of A and $50 per acre of B. He can spend at most $1840 on cultivation for these two crops. His profits are $400 per acre of A and $500 per acre of B. How many acres of each should he plant to maximize his profit?

 

2. Same as 1 but include the additional constraint: Each acre of A requires 40 labor hours and each acre of B requires 30 labor hours. He can use at most 1450 labor hours.

 

3. A, B, and C are sets in a universal set U. Describe each of the sets in terms of A, B, and C and represent each set with a shaded Venn diagram.

a){x| x is in at least one of A, B, and C} 

b) {x| x is in A and B but x is not in C}

c) {x| x is in A or B and x is not in C}  

 

4. A={x| x is an accounting major at TAMU} B={x| x is a junior accounting major at TAMU}  C={x| x is a junior at TAMU}  D={x| x is a psychology major at TAMU}

Which are true? (The notation must be correct for the statement to be true.)

 a) B   b)    c)    d)    e)   f) B= 

 

 

5. For A, B, C, and D as in 4, describe each of the following sets in words.

 

a)    b)   c)    d)   e)    f)

 

g). All the psychology majors and all the accounting majors are invited to a meeting. What set describes those invited?

 

6. 120 people were asked if they read or listen to music in their leisure time. 85 said they read. 65 said they listen to music. 10 said they do neither.

a) How many do both?     b) How many read but do not listen to music?   c) How many either read or listen to music but not both?

 

 

7. 100 children took a vision test and a hearing test. 50 passed the vision test. 60 passed the hearing test. 15 failed both tests.

 a) How many passed both tests?     b) How many passed the hearing test only?

 

 

 

 

8. Sixty students were asked if they belong to the theatre club, the Spanish club, or the art club.  From their responses it was found that:

13 belong to the art club only.

25 belong to the theatre club or Spanish but not the art club.

33 belong to exactly one of the three.

24 belong to the Spanish club only or the art club only.

20 belong to exactly two of the three.

32 belong to the art club.

22 belong to the theatre club but not all three.

 

Fill in a Venn diagram with the number in each region.

a) How many belong to none of the three?

b) How many belong to only one club?

 

 

9. A multiple choice test has 15 questions. Each question has 1 correct answer choice and 3 incorrect choices.

a)      How many ways can the test be answered?

b)      How many ways can the test be answered so that exactly one question is incorrect?

c)      How many ways can the test be answered so that exactly 13 questions are correct?

d)      How many ways can the test be answered so that exactly 10 questions are correct?

 

10. a) How many 8-letter words with no repeat letters can be made from 8 distinct letters?

      b) How many ways can a word of 4 letters be formed from 8 distinct letters if there are no repeats?

      c) How many 4-letter words can be made from 8 distinct letters if repeats are allowed?

       d) How many ways can 4 letters be chosen out of 8?

 

11.  How many ways can 12 jurors be chosen from 20 male and 30 female potential jurors:

a) no restriction is made.  b) there must be an equal number of men and women

 

12. There are 6 lawyers and 3 cases to be assigned lawyers. How many ways can this be done if:    a) one different lawyer is assigned to each case?

        b) each case gets one lawyer but the same lawyer can work on any number of cases?

 

13. There are 20 people in a group. They must choose a committee consisting of a president, vice president, secretary, treasurer and 10 regulars. How many ways can this be done?

 

14. A coin is tossed 5 times. How many ways can there be exactly two heads?

 

15. How many ways can you line up 6 couples so that spouses are next to each other?

 

16. Three presidents, 4 vice presidents, and 2 secretaries are to be in a photo. How many ways can they be arranged if people of the same rank must be kept together?

 

17. A room of children has 16 boys and 14 girls. How many ways can a group of  8 be chosen so there are 5 boys and 3 girls?

 

18. For each experiment, give a suitable sample space, S, and give n(S), and n(A) for the given event A.

a) Observe the sum of the top numbers when two dice are tossed. A is the event that there is at least one 4 or at least one 6.

 

b) Select 3 balls from a box containing 3 red, 5 blue and 7 yellow balls. A is the event exactly 1 blue or exactly two yellow are chosen.

 

19. E and F are two events in a probability space.

a) P(E)=0.2 and P(F)=0.4 and E and F are mutually exclusive. Find P(EF).

 

b) P(E)=0.3 and P(F)=0.8 . Could E and F be mutually exclusive? Why or why not?

 

20.       A survey asked 200 people to rate how well their school district is teaching their children. The responses were

Excellent    Good    Fair    Poor    No Opinion

       75         60         35       20         10

What is the probability that a randomly selected respondent

a)         rated the school district as fair, good or excellent?

b)         had an opinion?

c)         rated the school district as fair or poor?

d)         did not rate the school as excellent?

 

21. What is wrong here? In a raffle there are 500 tickets and 10 are winning tickets.

So a person who buys one ticket has a 1 in 50 chance of winning. Then if 50 people each buy one ticket one of them is sure to win.

 

 

22. P(E)=0.5, P(F)=0.6  P(E F)=0.9   Find

a) P( )     b) P()    c) P( )    

 

23. An urn contains 5 red, 4 blue and 3 yellow balls. Two are chosen at random. How many ways can each of the following be chosen?

a) at least one red?

b) at least one blue?

c) at least one red or at least one blue?(You can’t add answers to a and b here.)

d) two balls of the same color?

e) two balls of different colors?

 

Key:

  1. plant  8 acres of A, 32 acres of B for maximum profit of $19200. No leftover resources.
  2. same as 1 but there are 170 labor hours leftover.

 

  1. a)

 

  1. Only a is false.

 

  1. a) All the students who are majoring in accounting or psychology at TAMU.

      b) All the junior psychology majors at TAMU

      c) All the students who are juniors or are psychology majors at TAMU

     d)  Juniors at TAMU who are not psychology majors.

     e) Accounting majors at TAMU who are not juniors.

     f) All the students at TAMU who are juniors or are not psychology majors.

     g) AD

 

6. a) 40   b) 45   c) 70

 

7. a) 25   b) 35

 

8. a) 3   b) 33

 

9. a) 4     b) 45    c) 945    d) 729729

 

10. a) 8!=40320    b) 1680    c) 4096   d) 70

 

11. a) C(50,12)    b) C(20,6)C(30,6)

 

12. a) 120    b) 216

 

13. P(20,4)C(16,10) = 20! / (10!6!)

 

14. 10      15. 46080      16. 3!3!4!2!=1728

 

17. C(16,5)C(14,3)=1589952

 

18. a) S is all the ordered pairs of numbers 1, 2, 3, 4, 5, and 6 with repeats allowed. N(S)=36   n(A)=20

 

b) S is all the 3-element subsets of  { r1, r2, r3, b1, b2, b3, b4, b5, y1, y2, y3, y4, y5, y6, y7 }   n(S)=C(15,3)=455   n(A)=288

 

19. a) 0.6   b) no since 0.3 + 0.8 > 1         20. a) 0.85  b) 0.95     c)  0.275    d)  0.625

 

21. We cannot add without subtracting the intersections because the events are not mutually exclusive.

 

22. a) 0.2   b) 0.3   c) 0.1  

 

23.  a)  45    b) 38   c) 63   d) 19   e) 47