Math 166 Week-in-Review - © S. Nite 9/8/2012
WIR #2
Page 1 of 6
Week in Review #2 (1.1-1.4)
The union of sets A and B (A ∪ B) is the set of all elements that belong to either A or B or both.
A ∪ B = {x | x ∈ A or x ∈ B or both}
The intersection of sets A and B (A ∩ B) is the set of all elements in common with the sets A and
B. A ∩ B = {x | x ∈ A and x ∈ B}
Two sets are disjoint if they have no elements in common, that is, if A ∩ B = ∅.
If U is a universal set and is a subset of U, then the set of all elements in U that are not in A is
called the complement of A and is denoted A C . A C = {x | x ∈ U, x ∉ A}
If U is a universal set and A is a subset of U, then
b. ∅C = U
c. ( A C ) C = A
d. A ∪ A C = ∅
e. A ∩ A C = ∅
a. U C = ∅
Let U be a universal set. If A, B, and C are arbitrary subsets of U, then
Commutative laws for union & intersection A ∪ B = B ∪ A A ∩ B = B ∩ A
Associative laws
A ∪ (B ∪ C) = (A ∪ B) ∪ C A ∩ (B ∩ C) = A ∩ (B ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Distributive laws
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
De Morgan’s Laws
Let A and B be sets. Then, ( A ∪ B ) C = A C ∩ B C and ( A ∩ B ) C = A C ∪ B C
If A is a set, then n(A) denotes the number of elements in the set.
If A and B are disjoint sets, then n(A ∪ B) = n(A) +n(B).
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(A ∩ C) – n(B ∩ C) + n(A ∩ B ∩ C)
An experiment is an activity with observable results. Each repetition of an experiment is a trial.
The results of the experiment are called the outcomes. The set consisting of all possible sample
points of an experiment is the sample space. A sample space associated with an experiment that
has a finite number of possible outcomes (sample points) is called a finite sample space. A
subset of a sample space of an experiment is an event. A simple event is an event with a single
outcome. The empty set is the impossible event. An event that is same as the sample space is the
certainty event. E & F are mutually exclusive if E ∩ F = ∅.
Sample spaces in which the outcomes are equally likely are called uniform sample spaces. The
probability of an event is a number that lies between 0 and 1.
If S is a finite uniform sample space and E is an event, then the probability of E, is given by
Number of elements in E n( E )
P( E ) =
=
Number of elements in S n( S )
A probability distribution for an experiment is a table of probabilities for each of the simple
events. If an event E occurs m times in n trials, the ratio m/n is the relative frequency of the
event E after n repetitions. The relative frequency distribution is often referred to as an observed
or empirical probability distribution.
If E is an event of an experiment and E C
denotes the complement of E, then
P(EC) = 1 – P(E)
Union Rule: If E and F are any two events of an experiment, then
P(E ∪ F) = P(E) + P(F) – P(E ∩ F)
Math 166 Week-in-Review - © S. Nite 9/8/2012
WIR #2
Page 2 of 6
1. Given that n(AC ∩ B) = 60, n(A) = 85, n(BC) = 140, and n(U) = 205,
what is n(A ∪ B)?
U
A
B
2. Given that n(A ∩ B ∩ CC) = 30, n(A) = 125, n(A ∩ B ∩ C) = 15, n(C ∩ B) = 70,
n(U) = 250, n(B) = 100, n(C) = 140, and n(A ∩ C) = 65, find n(AC ∩ B ∩ C).
U
B
A
C
Math 166 Week-in-Review - © S. Nite 9/8/2012
WIR #2
Page 3 of 6
3. A survey of 400 people was conducted to see which of the following three
shows they watched: The Mentalist, NCIS, and CSI: Miami.
176 watched The Mentalist
227 watched NCIS
129 watched CSI: Miami
101 watched NCIS and CSI: Miami
123 watched The Mentalist and NCIS
84 watched The Mentalist and CSI: Miami
56 watched all three
U
N
M
C
a) Fill in the Venn diagram.
b) How many people watched exactly one of the three shows?
c) How many people did not watch NCIS or The Mentalist?
4. A jar contains eight blue marbles numbered 1 to 8 and fourteen green marbles
numbered 1 to 14. A marble is drawn at random from the jar. Find the probability
that
a) the marble is green.
b) the marble is odd.
c) the marble is green or odd.
Math 166 Week-in-Review - © S. Nite 9/8/2012
WIR #2
Page 4 of 6
5. An experiment consists of picking an integer from 1 to 20, inclusive.
a) Describe an appropriate sample space.
b) Describe the event E that the number picked was a multiple of 3.
c) Describe the event F that the number picked was odd.
d) Describe the event G that the number picked was a multiple of 4.
e) Describe the event H that the number picked was even or a multiple of 5.
f) Find the event GC ∩ (F ∪ E).
g) Which pairs of events, E, F, G, and H are mutually exclusive?
h) If the integer 18 were picked, which of the events E, F, G, and H occurred?
i) What is the probability of each of the events E, F, G, and H?
j) What is the probability of F ∪ G? F ∩ (HC ∩ E)? GC ∩ (F ∪ E)?
Math 166 Week-in-Review - © S. Nite 9/8/2012
WIR #2
Page 5 of 6
6. The odds of winning a certain carnival game are 3 to 70. What is the probability
of losing the game?
7. At a certain tent sale two days before the first Aggie game in the SEC, the
number of SEC Aggie shirts sold are given in the table below:
Size
Number
Small
Medium
Large
X-Large
XX-Large
235
350
500
450
385
a) Write the empirical probability distribution table.
b) What is the empirical probability that a person who bought one of the shirts
chose a medium or an X-Large?
8. Shade the portion of the figure that represents the set (A ∪ CC) ∩ B.
U
B
A
C
Math 166 Week-in-Review - © S. Nite 9/8/2012
WIR #2
Page 6 of 6
9. Five quarters, two dimes, and three nickels are in a piggy bank. Two coins are
drawn at the same time, and the total dollar amount is recorded.
a) Write the sample space.
b) Describe the event that the total dollar amount is less than $0.17.
10. A sample of 100 students were asked their major and whether they were
enrolled in a math course. The results are in the table below.
In Math Course
Math Major
Engineering Major
Biology Major
Total
28
31
17
76
Not in Math
Course
0
5
19
24
A student is selected at random from the sample. What is the probability that the
student
a) was not enrolled in a math course?
b) was an engineering major not enrolled in a math course?
c) was a biology major or a math major enrolled in a math course?