Spring 2016 Math 166
4. Two fair spinners each numbered 1-5
are spun and the numbers pointed to
Final Exam Week in Review
are observed.
courtesy: David J. Manuel
(a) In how many ways can at least one 5 be
spun?
(covering all Sections:
L.1-L.2; 1.1-3.4; F.1-F.4;
(b) What is the probability that the sum of
4.3-5.3; M.1-M.3; G.1-G.2)
the numbers is 6?
(c) What is the probability that a 2 is spun
p be the statement Houston is the
capital of Texas. and q be the state-
1. Let
or the sum of the numbers is no more
than 5?
ment Alaska is the largest state in the
USA.
5. Let
the capital of Texas and Alaska is not
ing probability distribution:
the largest state in the USA.
(b) Write in words the statement
(c) Given
p
is false and
q
s = {1, 2, 3, 4, 5, 6} be the sample
space of an experiment with the follow-
(a) Write the symbolic form of Houston is
∼p∨q
Outcome
1
Probability
6
16
2
3
4
?
2
16
2
16
5
6
?
1
16
is true, nd the
truth values of the statements in (a) and
(b).
2. Given
(a) Let
a
universal
set
U
Given
P (A) =
nd the missing probability values.
=
{a, b, c, d, e, f, g, h, i, j}
with
sets
A
=
{a, c, e, g, i},
B = {b, d, f, h, j}, and C =
{a, b, d, e, h, i}:
(a) Find the set
A = {1, 2, 3}.
11
,
16
(b) Find the mean, median, mode, standard
deviation, and variance of the distribution above.
6. In a survey of voters, 80% identied
c
(A ∪ B) ∪ (B ∩ C c )
as Democrats. Of those who identied
(b) Give two sets which are disjoint with the
as Democrats, 85% supported stricter
set in (a).
safety equipment on guns, 9% opposed
stricter safety measures, and 6% were
3. In a survey of 163 students on pet own-
undecided.
ership:
Of those who identied
as Republicans, 65% supported stricter
26 owned a cat, dog, and sh
37 owned a dog and a sh
safety equipment, 25% opposed, and
11 owned a dog and a cat, but not a
10% were undecided.
sh
18 owned a sh but not a cat
78 owned a dog
(a) Draw a tree diagram for this experiment.
78 did not own a sh
71 owned exactly one of these types of
(b) What is the probability that a person
pets
from the survey identied as Democrat
Fill in a Venn Diagram using this in-
given that they opposed stricter safety
formation.
equipment on guns?
1
7. Four married couples attend Captain
11. Historical grades are the MATH 166 -
America: Civil War and sit in one row.
nal exam are normally distributed with
a mean of 76 and a standard deviation
of 12.
(a) In how many dierent ways can they be
seated if there are no restrictions?
(a) What is the probability of making a 90
(b) In how many dierent ways can they be
or better?
seated if each married couple wants to
sit together?
(b) If the professor wants at most 10% of the
students to fail the nal, what should
the minimum passing score be (to the
8. Eight
cards are selected at random
nearest tenth)?
without replacement from a standard
52-card deck.
12. An account earns 4% per year com-
pounded quarterly.
(a) How many dierent hands have at least
How many years
does it take for the balance to grow
one spade drawn?
from $2,000 to $3,200?
(b) What is the probability that exactly four
of the cards are of one suit and exactly
13. Suppose instead of purchasing the in-
four are from a dierent suit?
surance in #10, the employee decides
to invest that money themselves each
year in an account earning 5% per year
9. Suppose 12% of all used PS-4 games
compounded annually. How long will it
are defective. If 53 are sold in a given
take for the account to reach $100,000?
week...
14. You purchase a new home for $360,000,
(a) What is the probability that more than
paying 5% down and nancing the rest
7 of them will be returned?
at 3.9% per year compounded monthly
(b) What is the probability that at least 5
for 30 years. What is the equity in your
but fewer than 12 will be returned?
home if you decide to move in 6 years?
10. An
Accidental
icy.
15. Unnished furniture produces end ta-
employee wishes to purchase an
Death
or
Disability
bles, coee tables, and chairs.
pol-
Each
end table requires 1 hour of cutting, 1
The policy pays the beneciary
hour of gluing, and 2 hours of paint-
$100,000 in the event that the em-
ing. Each coee table requires 2 hours
ployee dies during the next year and
of cutting, 1 hour of gluing, and 3
$50,000 in the event that the employee
hours of painting. Each chair requires
2 hours of cutting, 7 hours of gluing,
becomes disabled during the next year.
and 9 hours of painting. How many of
The probability of each of these events
each type of item can they produce if
is .001 and .012 respectively. What is
they have 40 hours of cutting, 38 hours
the minimum the employee can expect
of gluing, and 78 hours of painting la-
to pay for this policy?
bor available?
2
16. Let
A
1 2
5 5
=
11 2x
6y 11
(b)
and
B
not, determine the optimal strategies
=
and the expected value of the game.
. Find
x
and
(a)
9 8
A − 2B =
15 1
36 47
AB =
43 31
y

0
 −2
−3
given...
17. Each summer your family either goes
to the beach, goes to the lake, or stays
at home.
If you go to the beach one
year, there is a 60% chance you go to
the beach the next year, 30% you go
to the lake, and 10% you stay home.
If you go to the lake one year, there is
a 40% you go to the beach next year,
20% chance you go to the lake, and
40% chance you stay home. If you stay
home one year, there is a 15% you go
to the beach the next year and an 85%
chance you go to the lake.
(a) If you stay home this year, what is the
probability that you will go to the lake
in four years?
(b) Find the steady-state distribution.
18. The following matrix is a transition
matrix for an absorbing Markov process. Find the limiting matrix.


1 0 0.2 0.15
 0 1 0.3
0.3 


 0 0 0.15 0.25 
0 0 0.35 0.3
19. Determine if the matrix below repre-
sents a strictly-determined game, and
if so, nd the value of the game.

1
 −5
4
−2
0
1
0
−6
1

3
8 
2
20. Determine if the matrix below repre-
sents a strictly-determined game, and
if so, nd the value of the game.
If
3

−1 1
1 3 
−1 0