MA 114 Introduction to Finite Mathematics
Final Exam Practice Run Solution
June 26, 2010
Instructions: Show all work and justify all your steps relevant to the solution of each problem. No texts,
notes, or other aids are permitted. Calculators are permitted. Please turn off your cellphones. You have 180
minutes to complete this exam.
1. Go back through your old exams and re-work all problems. Pay close attention to any for which
you did not receive full credit! (But don’t ignore the others, either)
Note. The sample questions below do not span the entire set of possible topics. You should be
familiar with everything covered in previous exams and homeworks. The questions below are to
help you self-assess your progress.
2. A mathematics instructor selects one problem at random from the homeworks and places it, verbatim,
on the final exam. Determine the probability it looks familiar because you did all the homework.
3. A mathematics instructor selects one problem at random from the previous exams and places it, verbatim, on the final exam. Determine the probability you know how to answer it because you took
problem 1 seriously.
4. Suppose in a group of 200 people, 30 are swimmers, 50 are joggers, and 50 are tennis players. Assume
also 18 swim and jog, 25 jog and play tennis, 15 swim and play tennis, and 10 do all three activities.
What is the probability that a randomly chosen person plays tennis or jogs, but does not swim? What
is the probability that a randomly chosen person doesn’t participate in any of the activities?
(*) The answers are 52/200, 118/200, respectively. Draw a Venn diagram to compute the
number of people in each category. Since the probability is uniform, you can compute it by
counting the number of elements in the event space and sample space.
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5. Kathy has 3 red barrettes and 2 green barrettes in a box. She pulls out barrettes one at a time without
replacement until she has two of the same color. Draw a probability tree diagram illustrating the
outcomes of the scenario. Determine the probability she draws 2 green barrettes. Determine the
probability she has to pull out 3 barrettes given she gets 2 green barrettes. What is the expected
number of green barrettes she chooses?
(*) Diagram below.
6. Jim is playing a carnival game. On average, the probability he wins a game is .125. Assume he plays
6 times and the games are independent. Determine the probability he wins at least 2 games, and
determine the probability he wins at least 2 games, given he wins the first game.
(*) P(wins at least twice) = 1−(P(wins 0)+P(wins 1)) = 1−(C(6, 0)( 87 )6 +C(6, 1)( 18 )1 ( 78 )5 )
≈ .1665
P(wins at least twice|wins first) = P(wins at least once out of 5) = 1 − C(5, 0)( 78 )5 ≈ .4871
7. Determine all solutions to the system
{4x + 2y − z = −1, y + z = 0, x + 3z = 1}
(*) x = 0, y = − 31 , z =
1
3
8. Determine all solutions to the system
{x + y + z = 0, y − z = 0, x + 2y = 0}
(*) Augmented matrix is

1 0
 0 1
0 0
2
−1
0
x = −2z, y = z.
2

0
0 
0
9. How many house designs are possible if a contractor offers 4 choices of roofs, 3 choices of windows,
and 6 choices of brick designs.
(*) C(4, 1) × C(3, 1) × C(6, 1)
10. How many ways are there to arrange the letters in the word “HANNAH”?
(*)
6!
2!2!2!
11. A student takes an exam with 2 questions. Each question has choices a,b,c. Write down the sample
space for this experiment.
(*) S = {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)}
12. Lisa has 7 green and 6 orange shirts. If she picks 3 of the shirts randomly to stuff in her suitcase for a
trip, what is the probability that all 3 shirts selected will be green.
(*)
C(7,3)×C(6,0)
C(13,3)
13. 4 boys and 2 girls have purchased tickets for seats together in a movie theatre. They seat themselves
randomly. Determine the conditional probability the boys sit together if you know the girls sit together.
(*) Illustration below.
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14. A company has stores in Denver and Cincinnati and sells items to men, women, and children. Sometimes items that don’t sell well are transferred from one store to another. Each week at the Denver
store, 20% of the items are sold to men, 10% are sold to women, 10% are sold to children, 20% are
shipped to Cincinnati, and the remaining 40% are kept in Denver. The Cincinnati store sells 10% to
men, 20% to women, 30% to children, ships 10% to Denver, and keeps the remaining 30%. Treat this
as a 5 state Markov chain. Give the transition matrix. Is it regular? If so, what is the steady-state
matrix?
(*) Matrix.


1
0
0
0
0
 0
1
0
0
0 


 0
0
1
0
0 


 0.1 0.2 0.3 0.3 0.1 
0.2 0.1 0.1 0.2 0.4
To determine if its regular, we can first recall a previous example. Remember when we
squared the transition matrix below
0 1
1
3
2
3
we determined it was regular. Conceptually this should make sense in the following way.
If we are in state 1, we have to move to state 2. Hence, the probability of going from state
1 to state 1 in one step is zero. However, we can go to state 2 in the first step, then back to
state 1 in the second step. So the probability of going from state 1 to state 1 in two steps is
non-zero. The matrix below is different.
1 0
1
3
2
3
Once we are in state 1, there is no budging - no matter how many times we repeat the
experiment. So the (1, 2) entry will always be zero no matter what the power of the matrix
is. This is the scenario in the bigger matrix above.
Since the matrix is not regular, there is no steady state matrix.
Note. the remaining parts of the problem have been erased. We did not end up making it
this far into Markov chains.
15. Given events A, B, C with probabilities P(A) = .3, P(Bc ) = .45, P(C) = .9, P(B ∩ C) = .2, P(A ∩ B) =
.165, find P(B|C). Are the events B and C independent? Are the events A and B independent? Find
P(A ∪ B).
(*) P(B|C) =
P(B∩C)
P(C)
=
2
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P(B ∩ C) 6= P(B) × P(C). They are not independent.
P(A ∩ B) = P(A) × P(B). They are independent.
P(A ∪ B) = P(A) + P(B) + P(A ∩ B) = .3 + .55 − .165.
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16. A small business makes 3-speed and 10-speed bicycles at two different factories. Factory A produces
16 3-speed and 20 10-speed bikes in one day while factory B produces 12 3-speed and 20 10-speed
bikes daily. It costs $10000/day to operate factory A and $800/day to operate factory B. An order
for 96 3-speed bikes and 140 10-speed bikes has just arrived. How many days should each factory
be operated in order to fill this order at minimum cost? What is the minimum cost? Solve using
Geometric Linear Programming. Is it a standard problem?
(*) A minimization problem is not a standard problem.
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17. A lake is to be stocked with trout and salmon. Two foods, A and B, are available in the lake. Each
trout requires 2 units of A and 1 unit of B per day while each salmon requires 1 unit of A and 3 units
of B per day. Only 500 units of A and 900 units of B are available daily. If each trout weighs 2 lbs. and
each salmon weighs 4 lbs., how should the lake be stocked to maximize the weight of the fish which
can be supported by the lake? Solve using the Simplex Method.
(*) Let x denote number of trout. y denote number of salmon. Constraints are: (Food A)
2x + y ≤ 500. (Food B) x + 3y ≤ 900. (Objective) Maximize P = 2x + 4y.
Simplex Tableau

2
 1
−2
1 1 0 0
3 0 1 0
−4 0 0 1

500
900 
0
Column 2 is the pivot column and entry 2 is the pivot. (The (2, 2) position). After one pivot
operation the tableau is
 5

0 1 − 31 0 200
3
 1 1 0 1 0 300 
3
3
− 32 0 0 34 1 1200
We are not done yet. The first column is the pivot column. The pivot is in the (1, 1) position.
After the second pivot operation the tableau is


1 0 53 − 51 0 120
2
 0 1 −1
0 260 
5
5
2
6
0 0 5
1 1280
5
And we’re done. x = 120, y = 260, and P = 1280.
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