Discrete Math Final Exam Review- Spring 2013
Name__________________________________
For #’s 1-10, use the following set of Preference Schedules:
A
B
B
C
B
C
A
B
C
D
C
D
D
A
D
A
23
13
9
16
1. Which choice is the plurality winner?
2. What percentage of the first place votes did this winner receive?
3. Was the plurality winner a majority winner?
4. In a sequential runoff, which choice is eliminated first?
5. Which choice wins the sequential runoff?
6. Which choice would win the Condorcet method?
7. Which choice would win a 4-3-2-1 Borda count?
8. Which two choices would compete in a runoff?
9. Which choice would win a runoff?
10. In an approval vote, voters approve of their top two choices. Which choice wins the approval vote?
11. In a weighted voting situation, A has 7 votes, B has 4 votes, and C has 3 votes. What are the winning
coalitions and power index of each voter?
12. What is the saddle point for the following matrix, describing a game?
4
2
-2
-4
13. What type of game is exhibited in #12?
Questions #14 – 18 refer to the following game matrix:
4
-2
-1
2
14. Does this game have a saddle point?
15. What is a game like this called?
16. Suppose the row player chooses row 1 60% of the time. If this is the case, how often will the player
choose row 2?
17. Suppose the column player chooses column 1 30% of the time. If this is the case, how often will the
column player choose column 2?
18. Calculate the expected value for this game. Which player benefits from the game?
19. For the following game, decide the row player’s best strategy. FYI: The best strategy MAY be to use
a mix of strategies!
-4
2
4
-5
20. Which matrix below could be a Markov transition matrix?
A)
D)
.2
.3
B)
.8
.7
.1
.1
.8
.2
.5
.3
.4
.4
.2
.4
.6
1.2
-.2
C)
.1
.2
.7
.8
.1
.1
21. For the Markov transition matrix from above, find a stable state vector to describe the long-term
behavior of the Markov chain. You may use any initial condition that you like, as long as the
probabilities in this condition add to 1.0!
22. There are 8 finalists in the 100-meter hurdles. In how many ways can the gold, silver, and
bronze medals be awarded?
23. In how many ways can the letters in the word peninsulas be arranged?
24. In how many ways can 10 kindergarteners be arranged in a line to enter into a school cafeteria?
25. Four men and four women are to be selected from a group consisting of 10 men and 11
women. In how many ways can this be done?
26. Currently, Volkswagen offers four trim models for the 2011 Jetta. For each trim, you may
choose automatic or manual transmission. You may also chose from six different colors. How
many different VW Jettas are there for 2011?
27. How many different four student groups can be made from a group of 12 students?
28. In how many ways can the letters in the word stardom be arranged?
29. In an arithmetic sequence, if the first term is -2 and the common difference is 5, find the 12th
term.
30. What is the common ratio for the geometric sequence:
4, 2, 0.5, 0.25, 0.125, …
31. Find a formula tn = t1 + (n – 1)∙d to describe the following arithmetic sequence: 50, 48, 46, 44,
42, …
32. If the first term of an arithmetic series is 3, the common difference is 0.5, and the last term is
7.5, what is the sum of the arithmetic series? You need to use
Sn 
n
t1  t n  to find the
2
sum, but you have to figure out how many terms, n, are in the sequence first using the general
Arithmetic sequence formula: tn = t1 + (n – 1)∙d
33. Find two terms for an arithmetic sequence between 6 and 45. You can use tn = t1 + (n – 1)∙d to
find d!!
34. What is the 4th term for a geometric sequence for which t1 = 5 and r = 3? Use tn = t1∙rn – 1?
35. What is the sum of the first 10 terms of the geometric sequence described in #34? You may
find it helpful to use the formula: S n 


t1 1  r n
.
1 r
36. The sequence: 5, 10, 27, 62, 121, 210, 335, … is generated by a polynomial function. Use the
method of common differences to find the degree of and an equation for the polynomial
function that generates the sequence.
37. A sequence described by the formula: tn = a∙tn – 1 +b is called?
38. Give a recursive formula (in terms of tn and tn – 1) for the sequence: 5, 3, 1, -1, -3, -5, …
39. If tn = a∙tn – 1 – 3, and 6 is the fixed point, what is the value of a?
40. Find a formula of the form tn = a∙tn – 1 +b to describe the sequence: 5, 13, 29, 61, 125, 253, …
For #’s 41 – 43, use the following information:
An initial population for a given species is: [10 6 5 3 0 0] and birth/survival rates are:
Age (yrs)
0–2
2–4
4–6
6–8
8 – 10
10 - 12
BR
0
0.4
0.9
0.8
0.5
0.1
SR
0.6
0.8
0.7
0.5
0.4
0
41. After 6 years, how many animals will be on the 6 – 8 year age group?
42. What is the population total after 5 cycles?
43. What is the long-term growth rate for this species?
T
M
44. For the Leontief consumption matrix: T 0.08 0.15 , find a matrix to describe the amount
M 0.11 0.10
available for outside sale (D) if the production amounts are $30,000 in T and $20,000 in M. You
should use the matrix equation: D = P – CP.
45. How much should each sector produce (from #44) in order to meet a consumer demand of T =
M = $50,000? Use the matrix equation: P = (I – C)-1·D.
46. How many vertices and how many edges are there in a K2,3 graph?
47. What is the weight for a minimum spanning tree described by: V = {A, B, C, D, E, F} and
E = {AB-1; AC-2; AF-3; BC-5; CD-2; DE-4; EF-3}?
48. For the adjacency matrix:
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
A
A B C
0 2 1
D
2
B
1
3
0
0
C
2
1
1
2
D
0
1
1
1
describing a directed graph, how many directed
edges are there from C to B?
For a graph described by: V = {A, B, C, D, E} and E = {AC, AD, AE, BD, BE, CD, CE}, answer the
following:
a. Is the graph bipartite?
b. Is the graph a tree?
c. What is the chromatic number?
d. Will the graph have an Euler Path, Euler Circuit, or neither?
At most, how many child vertices can any parent vertex have in a binary tree?
What is the difference between Hamiltonian and Euler circuits?
Evaluate the RPN: 5 6 + – 2 6 3 – x – .
How many edges will a K8 graph have? You may want to detect a pattern in order to answer
this. How many edges in a K1, a K2, a K3, a K4, etc…??
If states A, B, and C have populations 647, 247, and 105 respectively, and there are 100 seats to
be apportioned among them, what are the initial and final apportionments for the Hamilton
and Webster methods?
If a die is tossed, what is the probability that it will show an even number or a 7?
Jimbo has 6 striped ties, 8 pattern ties, and 5 solid color ties. If he selects two ties at the same
time, what is the probability that they are both solid color? Here, you are finding the number
of ways of selecting two solid color ties divided by the number of ways of selecting two ties
from the total amount of ties.
If Jill has 3 red marbles, 5 green marbles, and 2 blue marbles, what is the probability that she
selects one of each color in a row if she makes each selection without replacement?
Give the same situation as above, what is the probability that, if Jill selects one marble, it is red
or green?
The likelihood that Ralph makes a free throw is 4/5. What is the probability that he will make
exactly 3 out of his next 6 free throws? Hint: binomialpdf(
A class has 12 seniors and 14 juniors. Five of the seniors are female and 8 of the juniors are
female. If a randomly chosen person is a junior, what is the probability that the person is a
male?
Four coins are tossed in the air, what is the probability that exactly three heads will come up?
If a card is drawn from a random deck of cards, what is the probability that it is a heart or a
jack?
Use the table below concerning high school students who skip/don’t skip and their GPA’s to answer
questions 63 – 65:
<2.0
2.0 – 3.0
>3.0
Total
Consistently Skip
80
25
5
110
Do Not Skip
175
450
265
890
Total
255
475
270
63. What is the probability that a student selected, who consistently skips, has a GPA > 3.0?
64. What is the probability that a randomly selected student has a GPA between 2.0 and 3.0 or
does not skip?
65. What is the probability that a student has a GPA that is less than 2.0, given that they do not
skip?
66. What is the difference between discrete and continuous division?
67. Skip, Vern, and Mookie will equally divide an estate consisting of a cabin, a car, and a painting,
in addition to $20,000. Skip bids $60,000 for the cabin, $6,000 for the car, and $2,000 for the
painting. Vern bids $52,000 for the cabin, $4,500 for the car, and $2,000 for the painting.
Mookie bids $61,000 for the cabin, $5,000 for the car, and $2,500 for the painting. State each
person’s final settlement.
68. An estate consists of $50,000 and a car. Alice bids $20,000 for the car and Barb bids $15,000
for the car. What are the final settlements for the estate if Alice is to get 60% of it?
69. Draw a Venn Diagram to illustrate (A  B)’C.
For #’s 70-78, use the following information: U = {0,1,2,3,4,5}; M = {0,1,2,3,4}; N = {1,2,3,4,5}; and
P = {1,2,4}. Write down the results of the indicated set operations.
70.
71.
72.
73.
74.
75.
76.
77.
78.
79.
M’ __________________
M  N __________________
P  M ______________
P’  M ________________
N’  P’ ________________
(M  N)  P’ ______________
(M’  P’)’ _______________
True or False? M and N are disjoint. If false, then what is their relationship?
True or False? P is a subset of M. If false, then what is their relationship?
Represent the coefficients for the following system of equations as a 2x2 matrix. Label it
Matrix A. What would -3·A21 be?
4x – 5y = 9
-2x + y = 10
80. Suppose a SRS of size 5 was used to find students to fill out a survey from a group of 36 students. If
the students were numbered 01-36, which students will be selected? Use the random number table
beginning at line 111.
81. A couple wishes to have a boy. They decided that they will have children until they either have a
boy, or have 5 children, whichever comes first. Conduct 15 simulations of this situation. Use your
Random Number Table beginning at line 133. Remember, a simulation of this experiment ends when
the couple either has a boy, or has 5 children. What is the probability that the couple have a boy within
the first five child births?
82. Suppose that in a particular year, the American League baseball team is considered to have a 60%
chance of beating the National League team in a given World Series game. To win the World Series, a
team must win 4 out of 7 games in the series. Conduct 15 simulations of possible World Series games.
Use the Random Number table beginning at line 140. Remember, the series is over once a team has
four wins. What is the probability that the American League team wins the series? What is the
probability that the National League team wins the series?
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Assignments for the Review:
05/24/2013- #1 – 21 & Extra Credit Project (if you choose to do it)
05/28/2013- #22 – 40 & Extra Credit Project (if you choose to do
it)
05/29/2013- #41 – 69 & Extra Credit Project (if you choose to do
it)
05/30/2013- #70 – 82 & Extra Credit Project (if you choose to do
it)