Math 166 Final Review with key:
- Set up and solve a system of equations to answer the following:
A total of $100,000 was invested in a money market, gold and a stock fund. The interest rates earned on the money market, gold and stock fund were 5%, 6% and 10% respectively. The total interest earned for the year was $8600. There was three times as much money in stocks as in gold. How much was invested in each fund?
2. 
a) Compute 2A+3B b) Compute 
3. A manufacturer has 2 kinds of products at 2 different locations. Matrix A shows the number in each location at the beginning of December. Matrix B shows the number sold in December. If he expects to sell 20% of his remaining stock in January, what matrix shows how much he will have left at the beginning of February?
4. Find the product AB for 
5. Two salesmen travel in 2 different cars. Salesman 1 uses a regular civic which has an average cost of 4 cents per mile in the city and 3 cents per mile in the country. Salesman 2 uses a civic hybrid which has an average cost of 2.5 cents per mile in the city and 1.8 cents per mile in the country. They both average 150 miles in the city and 50 miles in the country per day. What matrix product shows the average daily cost for each car.
6. A company produces 2 kinds of candy. Chocolate chunk uses 3 lb of butter, 1 lb of chocolate and 2 lbs of sugar per batch. Fudge uses 2 lbs of butter, 2 lbs of chocolate, and 1.5 lbs of sugar per batch. Butter costs $3 per lb, chocolate costs $2.50 per lb and sugar costs $1 per lb. They have an order for 5 batches of chocolate chunk and 6 batches of fudge. What matrix product gives the total cost of the order?
7. Solve each system of equations or state there is no solution. If infinite, give 2 particular solutions.
a) x+2y+3z=10, x – y - 2z=15, 5x - y=35
b) x+y+z=1, 2x – 3y+2z=0, 4x – y+4z=5
c)x+2y – z=3, 2x – 3y+z=5, x – 5y+2z=2
8. Which set of equations in 7 can be solved using an
inverse matrix? Set up this system as AX=B. Find 
.
9. Pivot the matrix about the specified entry.
a) 
row 1 col. 1 b) 
row 2 col. 2
10. An economy is based on three sectors, agriculture, energy and manufacturing.
Production of a dollar’s worth of agriculture requires an input of $0.20 from the agriculture sector, $0.40 from energy and none from manufacturing.
Production of a dollar’s worth of energy requires no agricultural input, an input of $0.20 from energy and $0.40 from manufacturing.
Production of a dollar’s worth of manufacturing requires $0.10 of agriculture, $0.10 of energy and $0.30 of manufacturing.
a) Find the output from each sector that is needed to meet a demand of $20 billion for agriculture, $10 billion for energy and $30 billion for manufacturing.
b) How much of each sector is consumed by producing the amounts found in a)?
11. U is the universal set. A, B, and C are subsets of U. Describe with unions, intersections and complements:
a) {x in U: x is in at least one of A, B, and C}
b) {x in U: x is in all of A, B, and C}
c) {x in U: x is in A and B but x is not C}
d) {x in U: x is in A or B but x is not in C}
e) {x in U: x is not in A or B but x is in C}
12. To get a flu shot, anyone under 60 must have a special condition. Anyone else can get a flu shot. A is the set of people under 60. B is the set of people with a special condition.
a) Describe in words the set of people who can get a flu shot.
b) Describe the set in a) with unions, intersections and complements.
c) Describe with unions, intersections and complements those who cannot get a flu shot.
13. Seven plays are nominated for three different prizes. How many ways can the prizes be awarded if:
a) no play gets more than one prize?
b) any play can get any number of prizes?
c) a single play can get at most 2 prizes?
14. How many distinguishable arrangements of 5 red, 4 blue and 3 yellow balls are there?
Think of
15. How many ways can 1 first prize, 5 second prizes, and 10 third prizes be awarded among 70 people if no-one gets more than one prize?
16. A legal firm has 5 new cases and 20 lawyers. How many ways can they assign lawyers to cases if
a) each case needs one lawyer and any lawyer can work on any number of cases?
b) each case needs one lawyer and no lawyer can work on more than one case?
c) one of the cases needs two lawyers, one needs four and the other three each need three assuming no lawyer works on more than one case.
17. Twelve athletes are to line up for a photo. There are 5 football players, 3 swimmers and 4 runners. How many ways can they line up if athletes of the same sport must be kept together?
18. A box contains 50 apples. 17 are honeycrisp, 20 are gala, and 13 are cameo. 5 apples are chosen at random. What is the probability of getting:
a) at least one honeycrisp?
b) at least 2 honeycrisp?
c) all of the same type?
d) exactly 2 honeycrisp or exactly 3 gala?
19. A multiple choice questionnaire has 15 questions which can be answered SA, A, N, D, or SD. If questions were answered randomly what is the probability that:
a) exactly 3 would be answered A or SA?
b) between 1 and 5 inclusive are answered A or SA? (Think binomial)
c) If X is the number of questions answered A or SA, what are the mean and standard deviation of X?
20. A store survey asked 150 people to rate the sales service on a scale of 0=awful to 5=excellent. The results were
rating 5 4 3 2 1 0
# of respondents 55 40 25 15 5 10
If X is a randomly selected rating find:
a) the mean, median, mode, standard deviation and variance of X.
b) P(X>3)
21. P(E)=0.7
P(F)=0.8 P(
)=0.06 a) Find
P(
). b) Are E
and F independent?
22. Six people choose from 12 varieties of trees to plant. What is the probability :
a) at least 2 people will choose the same variety?
b) they will all choose the same variety?
23. A disease has a 15% rate of occurrence. A test for the disease is negative in 10% of people who have the disease and positive in 12% of people who do not have the disease. Let + be the event the test is positive and D the event the person has the disease. Find the probability that
a) a randomly selected person gets a positive test.
b) a randomly selected person gets a positive test and has the disease.
c) a person who gets a positive test actually has the disease.
If the test is positive, a second more accurate test is done. This test is negative in only 4% of people who have the disease and is positive in only 6% of people who do not. Find the probability that
d) a person who does not have the disease gets positive results on both tests.
e) a person who gets positive results on both tests actually has the disease.
24. A fictitious study tested to see if caffeine could improve memory. 100 people drank coffee and were asked to memorize a list. 100 people consumed no caffeine and memorized the same list. 110 people remembered at least 90% of the list. 75 of the
people who drank coffee remembered at least 90% of the list. Let C represent those who drank coffee and M those who remembered at least 90% of the list.
a) According to this study, are C and M independent? Why or why not?
b) Does the study show coffee helps memory? Why or why not?
c) What is the probability someone who did not drink coffee remembered at least 90% of the list?
25. a) The odds in favor of a certain team winning a game are 3:4. What is the probability this team will not win?
b) The chance of rain is 40%. What are the odds in favor of rain?
26. The events that security checkpoints 1, 2 and 3 will detect an intruder are independent. The probabilities that systems 1, 2 and 3 will let an intruder pass through are 0.08, 0.07, and 0.06 respectively. What is the probability an intruder
a) will be detected by at least one checkpoint?
b) will be detected by checkpoints 1 or 2, possibly both?
27. A city has 3 hospitals A, B, and C. 40% of new babies in the year were born at hospital A and had an average weight of 7.2 lbs. 35% were born at hospital B and had an average weight of 6.9 lbs. 25% were born at hospital C and had an average weight of 6.5 lbs. What is the average weight of babies born in the city hospitals that year?
28. A stock is watched for 14 days with the following results:
price/share $45 $47 $49 $50 $52 $53
# days 1 3 2 2 2 4
Find the mean, median, mode, standard deviation and variance of the price per share.
In another 14 day period the prices were: $45 47 49 50 52 53
# days 4 2 1 1 3 3
Did the standard deviation increase, decrease or stay the same and why?
29. A student takes 14 credits and receives the following grades.
Grade A B C D F
# credits 3 4 4 3 0
a) Find the students gpa for the 14 credits.
b) If he previously had a gpa of 3.0 for 70 credits, find the new gpa.
30. A set of exam grades is normally distributed with a mean of 75 and a standard deviation of 12.
a) What is the probability a score is between 70 and 80 rounded to four decimal places?
b) Find a so the probability a randomly selected score is less than a is .7.
c) Find b so the probability that a randomly selected score is greater than b is .7.
d) Find the probability that a randomly selected score is better than or equal to 80 and round to 2 decimal places.
Ten students are selected at random. X is the number who score better than 80. Then X~B(10,p) where p is the answer to d.
e) What is the probability at least three will score better than 80?
f) What is the expected number to score better than 80?
31. The length, L, of a randomly selected baby born at a certain hospital is normally distributed with a mean of 20 inches and a standard deviation of 0.75 inches.
a) Find the probability L is between 19 and 21 inches.
Fifty babies are selected at random and X is the number who are between 19 and 21 inches long.
b) What is the distribution of X?
c) Find the mean and standard deviation of X.
d) What is the probability that X is at least 40?
e) Find P(40<X<42).
f) Find the normal approximation to the probability in e.
32. A professor wants to give A’s to the top ten percent of his class and B’s to the next 15%. If the scores are normally distributed with a mean of 70 and a standard deviation of 18, what should be the A and B cutoff scores?
33. X is a normally distributed random variable mean 80 and unknown standard deviation. If P(X>65) = .73, find each of the following probabilities.
a) P(X<65) b) P(X< 95) c) P(80< X < 95)
34. The weights of babies at a hospital are normally distributed with mean 7 lbs and standard deviation 0.5 lb. Find a weight, W, so that
a) 40% of babies weigh more than W.
b) 30% of babies weigh less than W.
35. A fair coin and a fair 4-sided die are tossed.
a) Describe a sample space for this experiment for which the probability distribution is uniform.
The random variable, X, is 1+the top number on the die if the coin lands heads and X is the top number on the die if the coin lands tails.
b) Describe the events (X=1) and (X=2).
c) Write the probability distribution for X.
d) Write the distribution of X if the coin is weighted so that the probability of heads is 0.4.
36. A person tosses a penny for which P(H) = 0.6, a nickel for which P(H) = 0.3, and a quarter for which P(H) = 0.2. X is the number of heads.
a) Write the distribution of X.
b) What is E(X)?
c) If a person wins $5 for each head, what is his expected winning?
d) If the person gets to keep 100 times the value of the coin for any coin that lands heads, what is his expected winning?
For d, you cannot use X. You have to make a tree.
37. Two coins are chosen from a set of 10 nickels and 5 dimes. X is the total value of the coins selected. Write the distribution of X.
38. A family had 5 children. The probability that a child in this family would have blue eyes is 0.5. What is the probability that the 2nd child had blue eyes if it is known that exactly 3 children had blue eyes?
39. Redo #36 if the probability of blue eyes is 0.25 for any child.
40. Q is the event that a person passes a certain quantitative test. V is the event that a person passes a verbal test. Q and V are independent. P(Q) = 0.7 and P(V) = 0.8.
The probabilities of graduating from a certain school are 0.4 for those who pass only the quantitative exam, 0.5 for those who pass only the verbal exam, 0.9 for those who pass both exams and 0.2 for those who pass neither exam. Make a tree diagram for this information. Remember that events for a given stage of the tree must be mutually exclusive.
What is the probability that someone who graduated passed both tests?
41. Find the steady states for the following 2-state regular Markov processes. Put your answers in fraction form.
a) The probability of remaining in state 1 is 0.35 and the probability of remaining in state 2 is 0.76.
b) The probability of moving from state 1 to state 2 is 0.56 and the probability of moving from state 2 to state 1 is 0.64.
c) The probability of moving to state 2 from state 1 is 0.68 and the probability of remaining in state 2 is 0.18.
Review word problems in chapter 9 sections 9.1, 9.2 and 9.4.
Be sure to also do the Finance Review.
Key:
- $10,000 in the money market, $22,500 in gold, $67,500 in stocks
2. a) 
b) 
3. 0.8(A-B)=
4. AB=
5. 
or 
6. 
or 
7. a)
x=10, y=15, z= -10 b) no solution c) 
t is any real number.
Some particular solutions are (20/7, 4/7, 1)
(19/7, 1/7, 0) (21/7, 1, 2)
8. Only a. The solution must be
unique. AX=B is 
9. a)
(1/3)R1, R2-2R1, R3+5R1 results in 
You could choose a
different sequence of row-ops with a different result in the second and third
columns..
10. a) $33 billion from agriculture, $37 billion from energy, $64 billion from manufacturing
b) $13 billion of agriculture, $27 billion of energy, and $34 billion of manufacturing
b) (1/2)R2,
R1-R2, R3-3R2 results in 
11. a) 
12. a) All people under 60 who have a special condition all people 60 or older.
b) 
c) 
13. a) P(7,3)=7x6x5=210 b) 343 c) 336
- 27,720
- 70C(69,5)C(64,10) or
- a)
3200000 b) P(20,5)=1860480 c)
- 3!5!4!3!
- a)
d) 
- a) 0.0634 b) 0.4027 c) 6 and 1.8974
- mean=3.6333 median=4 mode=5 standard deviation=1.4716 variance=2.1656 P(X>3)=0.8
- a) 0.56 b) yes
22. a) 
b) 
23. a) 0.237 b) 0.135 c) 0.5696 d) 0.0072 e) 0.9549
24. a) P(M|C)=0.75 P(M)=0.55 so they are not independent.
b) yes because P(M|C)>P(M)
c) 0.35
25. a) 4/7 b) 2:3
26. a) 0 .9996 b) 0.9944
27. 6.92 lbs
28. a) mean=50 median=50 mode=53 st.dev.=2.6458 var.=7.0003
b) st.dev increased because the prices farther from the mean have higher probabilities.
29. a) 2.5 b) 2.9167
30. a) 0.3231 b) 81.2928 or 81 c) 68.7072 d) 0.34 e) 0.7162 f) 3.4
31. a) 0.8176 b) binomial with n=50, p= 0.8176 c) mean=40.88 st.dev.= 2.7307 d) 0.7033 e) 0.4178 f) 0.4168
32. A cutoff score is 93, B cutoff is 82
33. a) 0.27 b) 0.73 c) 0.23
34. a) 7.1267 lbs b) 6.7378 lbs
35. a) { (H,1), (H,2), (H,3), (H,4), (T,1), (T,2), (T,3), (T,4)}
b) {(T,1)} {(H,1), (T,2)}
c) x 1 2 3 4 5
P(X=x) 1/8 1/4 1/4 1/4 1/8
d) x 1 2 3 4 5
P(X=x) 0.15 0.25 0.25 0.25 0.1
36. a) x 0 1 2 3
P(X=x) 0.224 0.488 0.252 0.036
b) mean is 1.1 c) $5.50 d) $7.10
37. x| 0 .10 0.15 0.20
p| 3/7 10/21 2/21
38. 0.6
39. 0.6
40. 0.7283
41. a) 