MA114-004 Final Review Game
Emese Lipcsey-Magyar
NCSU
December 4, 2013
Question 1
"
#
a b
State the formula for finding the inverse of a 2 × 2 matrix A =
.
c d
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December 4, 2013
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Question 2
Find the equation of the line passing through the point (1, 2) and
perpendicular to y = − 13 x + 5. Give your answer in standard form
(y = mx + b).
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December 4, 2013
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Question 3
Solve the following system


 x
−x


−x
Emese Lipcsey-Magyar (NCSU)
of equations using Gauss-Jordan elimination.
− 3y
+ 3y
+ 3y
+ 2z
− z
+ 2z
= 10
= −6
=
6
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December 4, 2013
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Question 4
A group of 180 business executives were surveyed to determine whether
they regularly read Fortune, Time, or Money magazines. The survey
showed that
75 read Fortune
70 read Time
55 read Money
45 read exactly two of the three magazines
25 read Fortune and Time
25 read Time and Money
5 read all three of the magazines.
Draw a Venn diagram to represent the given data.
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Question 5
An urn contains 6 red balls and 7 white balls. A sample of five balls is
selected at random.
a) How many different samples are possible?
b) How many samples contain all red balls?
c) How many samples contain at least one white ball?
d) How many samples contain either two or three red balls?
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December 4, 2013
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Question 6
A toy manufacturer makes lightweight balls for indoor play. The large
basketball uses 6 ounces of foam and 3 hours of labor and brings a profit
of $2. The football uses 3 ounces of foam and 4 hours of labor and brings
a profit of $3. The manufacturer has available 102 ounces of foam and
200 labor hours a day. Use the simplex method to determine the optimal
production schedule so as to maximize the profits. What is the maximum
daily profit?
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December 4, 2013
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Question 7
Find the first three terms of (x − 3y )10 .
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Question 8
Consider the following problem: minimize 5x + 7y subject to


2x + y ≥ 1



 x + 3y ≥ 2

x − 2y ≥ 3




x ≥ 0, y ≥ 0.
a) Determine the dual problem.
b) Use the simplex method to solve the dual problem and find the
solution to both the dual problem and the original problem.
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MA114-004 Final Review Game
December 4, 2013
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Question 9
A bag contains 5 blue, 4 yellow, and 3 green marbles. One marble is
removed, its color noted, and not replaced. A second marble is removed,
its color noted, and not replaced.
a) What is the probability that both marbles are blue?
b) What is the probability that the second marble is green, given that
the first marble is not green?
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December 4, 2013
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Question 10
A study of cigarette smokers determined that of the people who smoked
menthol cigarettes on a particular day, 10% smoked menthol the next day
and 90% smoked non-menthol. Of the people who smoked non-menthol
cigarettes on a particular day, the next day 30% smoked menthol and 70%
smoked non-menthol.
a) Set up a regular stochastic matrix that describes this transition.
b) In the long run, what percent of the people will be smoking
non-menthol cigarettes on a particular day?
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Question 11
Find the stable distribution for the

1

0
0
Emese Lipcsey-Magyar (NCSU)
following stochastic matrix.

.4 .1

.6 .4
0 .5
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December 4, 2013
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Question 12
Which of the given matrices are the payoff matrices of strictly determined
games? For those that are, determine the saddle point and the optimal
pure strategy for each of the players.
"
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"
#


1 −1 0
−1 −2
9 3
a)
b)


c) 6 3 2
0
3
2 6
2 −2 1
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Question 13
Consider the following payoff matrix for a not strictly determined game.
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4 3
2 4
a) Write down the linear programming problem that R has to solve in
order to find his optimal mixed strategy.
b) Solve the linear programming problem from part a) by graphing the
feasible set and evaluating the objective function at the vertices.
c) What is R’s optimal mixed strategy?
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Question 14
Find the inverse of A using Gauss-Jordan elimination.


1 2 −2


A = 1 1 1 
0 0 1
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