7.1 Introduction to Sets
Sets: A well-defined (clearly distinguishable) collection of objects in which it is possible to
determine if a given object is included in the collection.
We use {} to denote a set. We can describe a set by listing all objects in the set (roster notation).
We can also use set builder notation to describe a set and it has the form of {x: x describes the
characteristics of the object}. Lastly, we can use words notation by describing a set using words.
Elements/Members: the individual objects in the set (𝑁𝑜𝑡𝑎𝑡𝑖𝑜𝑛 ∶ ∈ )
Cardinality: n(A) or the number of elements in a finite set A.
Ex. 1 Find the cardinality of the set of all students enrolled in this section.
Notation of Sets:
a A 
a A 

0
0 
 
Ex. 2 Given the sets T  17,18,19,
answer the following:
, 25 and A  a : a is a positive even integer less than 21 ,
a. List the elements of A using set notation.
b. n(T)
c. True or False: 19  A
Equal Sets: If two sets have exactly the same elements. Note that the order doesn’t matter.
AB
{
}
{
}
Ex. 3 Given A= a,e,i,o,u , B = u,i,e,a,o , is 𝐴 = 𝐵?
Distinct (Different) Sets: 𝐴 ≠ 𝐵 if we can find an element in one set but not in the other set.
If two sets have same cardinalities, we say they are equivalent and denote it as 𝐴 ≈ 𝐵
Universal Set: U, a set that includes all the objects being discussed or under consideration.
Remark. The set of male students in Blinn college population (Universal set: The collection all
students, faculties, and staffs in Blinn College)
The set of male students in Texas population (Universal set: the collection of all Texans)
Subset: when all the elements of one set are included in another set (Notation:⊆ )
Notation:
AÍ C (Subset: same or small)
𝐴 ⊂ 𝐶 (Proper subset: strictly small)
  A, for all sets A
A  A, for all sets A
{
}
{
}
{
}
{ }
Ex. 4 Given A= 1,2,3,4,5,6 , B = 1,3,5 ,C = 3,5,1 and D = 5,3 then which of the
following are true?
a. B = C
b. B Í C
c.𝐵 ⊂ 𝐶
d. C Í B
e. 𝐷 ⊂ 𝐵
f. D Í B
g. 𝐵 ⊂ 𝐷
h. 4 ∈ 𝐴
i. {4,6} ∈ 𝐴
j. {2,3} ⊂ 𝐴
k. 1,5 ∈ 𝐶
l. D Í D
A set of k distinct elements had 2 k subsets.
Ex. 5 If B  {cow, moose, elephant} is a set, how many subsets of B are there? Then list all the
subsets.
We can use Venn Diagrams to organize sets and visualize subsets. There are certain definitions
associated with Venn Diagrams and sets.
Ways to create new sets from old sets
Complement of a Set: A is the set of
elements that are in U but not in A.
Intersection of Sets: A  B is the set of
elements in common with A and B.
U
U
A
A
B
A '   x x U , x  A
B
A  B   x x  A and x  B
Union of Sets: A  B is the set of all elements Disjoint Sets: When the intersection of two
sets is the empty set.
that belong to either A or B or both.
U
U
A
A
B
A  B   x x  A or x  B or both
𝐴∩𝐵 =∅
Let’s label various sections of the Venn Diagram below.
U
A
B
B
Ex. 6 Let U  1, 2,3, 4,5,6,7,8,9 ,
A  2, 4,6,8
B  1,3,5,7,9
C  1, 2,3, 4,5
D  5,6,7,8,9
Find each of the following.
1. C  B
2. A  D
3. A  B
4. AC  D
5.  A  C 
C
6. A  DC
C
7.  A  C   D
Remarks : (𝐴 ∪ 𝐵)𝑐 = 𝐴𝑐 ∩ 𝐵 𝑐 , (𝐴 ∩ 𝐵)𝑐 = 𝐴𝑐 ∪ 𝐵 𝑐 : Complement law
𝐶 ∪ (𝐴 ∩ 𝐵) = (𝐶 ∪ 𝐴) ∩ (𝐶 ∪ 𝐵), 𝐶 ∩ (𝐴 ∪ 𝐵) = (𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵): Distributive law
Prove this by using Venn diagram on your own
7.2 Applications of Venn Diagrams (Need some basic algebra)
Union Rule: 𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵) − 𝑛(𝐴 ∪ 𝐵)
U
A
B
Ex. 1 If n(A) = 15, n(B) = 30, and n(A  B) = 33, what is n(A  B)?
Ex. 2 If 𝑛(𝑈) = 33, 𝑛(𝐴 ∪ 𝐵) = 29, 𝑛(𝐴 ∩ 𝐵) = 5, and 𝑛(𝐵 ′ ) = 23, find 𝑛(𝐴′ ).
Ex. 3 Describe the identified sections in words and set operations. Let universal set 𝑈 = The set
of all people in Blinn College, W= the set of women, M= the set of married people, and S= the
set of students.
A=
Married
Women
B
C
A
F=
E
D
F
G
Students
H
H=
Venn Diagrams can be used to identify subsets in real-life situations.
Ex. 4 In a recent survey of 200 members of a local sports club, 100 members indicated that they
plan to attend the next Summer Olympic Games, 60 indicated they plan to attend the next Winter
Olympic Games, and 40 indicated they plan to attend both games. How many members plan to
attend:
a) At least one of the two games?
b) Exactly one of the games?
c) The Summer Olympic Games only?
d) None of the games?
Ex. 5 Draw a Venn Diagram to represent: An investigation of a number of automobile accidents
revealed:
18 accidents involved alcohol and excessive speed
26 accidents involved alcohol
12 accidents involved excessive speed but not alcohol
21 of the accidents involved neither alcohol nor excessive speed.
How many accidents were investigated?
Ex. 6 The National Resource Defense Council believes that by using technology properly, we
can cut U.S. global warming pollution by half. Three of the solutions proposed by the NDCR are
using energy-efficient appliances, driving energy-efficient cars, and using renewable energy
sources. Assume that you surveyed 100 members of Congress to determine which solutions they
favored funding and obtained the following results:
12 favored funding the increased use of renewable energy sources only
20 recommended funding both energy-efficient appliances and renewable energy sources
22 favored funding both energy-efficient cars and increased use of renewable energy sources
14 wanted to fund all three areas
From this information, determine the total number who favored increased funding for renewable
energy?
Ex. 7 Pandora.com surveyed a group of subscribers regarding which online music channels they
use on a regular basis. The following information summarizes their answers:
7 listened to rap, heavy metal, and alternative rock
10 listened to rap and heavy metal
13 listened to heavy metal and alternative rock
12 listened to rap and alternative rock
17 listened to rap
24 listened to heavy metal
22 listened to alternative rock
9 listened to none of these three channels
a. How many people were surveyed?
b. How many people listened to either rap or alternative?
c. How many listed to heavy metal only?
Ex. 8 250 college students were asked what they did last night. Completely fill in the diagram
below given the following information:
 169 watched TV
 16 went out to eat ONLY
 46 went out to eat and watched TV, but did not study
 132 did NOT study 
 18 went right to bed without doing any of these
 31 did all three things
 141 did at least 2 of these things.
Watched
TV
Ate Out
Studied
7.3 Introduction to Probability
Experiment: an activity or occurrence with an observable result
Trial: each repetition of an experiment
Sample Space: the set of all possible outcomes for an experiment
Events: a subset of a sample space ( A set of specific, prescribed, or wanted outcomes from
experiment)
Certain Event: when an event equals the sample space
Impossible Event: when the event is the empty set
Remark: If there are 𝑚 𝑤𝑎𝑦𝑠 of doing something and 𝑛 𝑤𝑎𝑦𝑠 of doing another, then there are
𝑚 ∙ 𝑛 𝑤𝑎𝑦𝑠 performing both actions. (Counting principle: a good way to find the number of
elements in sample space)
Ex. 1 Jerry tossed a coin, then spun a two-color spinner with red (R) and green (G) sections, and
then roled a 4-sided die (with numbers 1, 2, 3, and 4 on the faces).
a) What is the sample space for this experiment?
b) What is the event, E, that an even number is rolled?
c) What is the event, F, that a 5 is rolled?
Ex. 2 A committee of 2 turtles is selected from the 4 Ninja Turtles: Leonardo, Donatello,
Michelangelo and Raphael.
a) What is the event that Leonardo is on the committee?
b) What is the event that Michelangelo is NOT on the committee?
Operations (ways to create new from old): Suppose A, B are events for sample space S
A∩ 𝐵: the collection of all outcomes that are elements of both of A and B
(Event A ∩ 𝐵 𝑜𝑐𝑐𝑢𝑟𝑠 when both A and B occur)
A ∪ 𝐵:
𝐴′ ,
Recall: When talking about sets, we called (𝐴 ∩ 𝐵) = ∅ disjoint.
Mutually Exclusive Events: if the intersection of two events is null i.e., (𝐴 ∩ 𝐵) = ∅
Basic Probability formula: In case all outcomes are equally likely.
𝑃(𝐴) =
𝑛(𝐴)
𝑛(𝑆)
0  P ( E )  1 , S: sample space, A: Event, 𝑛(𝐴): the number of element in A,
𝑛(𝑆): the number of elements in S
Remark: 𝑃(𝑆) = 1, 𝑃(∅) = 0
Tree Diagram: A way of seeing what is going on ( Visual representation): Helpful when
working out probabilities for more than one event (sequential steps, repeated trials, or several
stages of classification)
Some Common Sample Spaces
Dice:
Cards:
Coins:
Ex. 3 What is the probability of rolling an odd number on a 12-sided die?
Ex. 4 What is the probability of drawing a number card greater than 7 from a standard deck of
cards?
Ex. 5 What is the probability of not drawing a face card from a standard deck of cards?
Ex. 6 A recent survey reported that 67% of Americans approve of human embryonic stem cell
research. If an American is selected at random, find the probability that he or she will disagree
or have no opinion on the issue.
Ex. 7 You select three digital picture frames from a production line to determine if they are
defective or not (under the assumption that outcomes are equally likely). How many elements
are in the sample space? What is the probability that exactly one frame is defective?
Ex. 8 An experiment consists of rolling an 8-sided die (numbered 1-8) and observing the number
that falls uppermost. Let E be the event that an odd number is rolled and F be the event that the
number showing is less than 5. Are the events E and F mutually exclusive?
Ex. 9 A marble is selected at random from a bag containing 3 blue, 4 red, 5 green, and 2 yellow
marbles. What is the probability that:
a. the marble is green?
b. the marble is red or yellow?
c. the marble is not blue?
d. the marble is blue and red?
e. the marble is blue, red, green or yellow?
7.4 Basic Probability
Union Rule for probability: P( A  B)  P( A) + P( B)  P( A  B)
Complement Rule: 𝑃(𝐴) = 1 − 𝑃(𝐴𝑐 )
Ex. 1 Two dice are rolled. The sample space is at right.
a. What is the event of rolling a sum of 4?
b. What is the probability of rolling a sum of 6?
c. What is the probability of rolling a 4 and a 5?
d. What is the probability of rolling a sum of 13?
e. What is the probability of not rolling a 6?
f. What is the probability of rolling a 3 or a sum of 7?
g. What is the probability that the sum is less than 5 or a 2 is rolled?
Ex. 2 A new medication produces headaches in 5% of the users, upset stomach in 15% and both
in 2%. Find the probability that:
a. at least one of these side effects occurs.
b. neither of these side effects occurs.
( )
Ex. 3 Let E and F be mutually exclusive events and suppose P(E)=0.1 and P F = 0.6 , find:
a. 𝑃(𝐸 ∩ 𝐹) =
b. 𝑃(𝐸 ∪ 𝐹) =
( )=
c. P E
c
d. 𝑃(𝐸 𝑐 ∩ 𝐹 𝑐 ) =
( )
Ex. 4 Let E and F be two events of an experiment with sample space S. Suppose P E = 0.6,
P( F ) = 0.4, and 𝑃(𝐸 ∩ 𝐹) = 0.15. Compute:
a. 𝑃(𝐸 ∪ 𝐹) =
( )=
b. P E
c
c. 𝑃(𝐸 ∩ 𝐹 𝑐 ) =
Ex. 5 You are drawing one card from a standard deck of 52 cards.
a) What is the probability you draw a heart or a face card?
b) What is the probability of drawing a spade or a diamond face card?
Odds: the odds in favor of E occurring are 𝑃(𝐸): 𝑃(𝐸 𝑐 ) 𝑜𝑟
the odds against E occurring are 𝑃(𝐸
𝑐 ):
(Reduce this in lowest terms)
Ex. 6 Two dice are rolled.
a) What are the odds of rolling a sum of 8?
b) What are the odds against rolling 2?
PE
1 P  E 

PE
P  EC 
, PE1
C
1 P  E  P  E 

, PE 0
𝑃(𝐸) 𝑜𝑟
PE
PE
Ex. 7 If the probability Courtney comes to class on time is 0.75, what are the odds against
Courtney coming to class on time?
Ex. 8 Find the odds in favor of drawing a face card from a deck of 52.
Ex. 9 During a game of Blackjack, three players are dealt two cards each, and the dealer has two
cards. No one has a card that is worth 10 (10 or a face card) or 11 (an ace).
a. What is the probability that the next player dealt a card would get a card worth 10 or 11?
b. What are the odds that the next player dealt a card would get a card worth 10 or 11?
If the odds in favor of an event, E, occurring are a to b, then the probability of E occurring is
P( E ) =
a
a+ b
Ex. 10 The odds that A&M will beat South Carolina this year are 9 to 7. What is the probability
that A&M wins?
Ex. 11 The odds in favor of the fire alarm going off are 9 to 3. What is the probability the fire
alarm will NOT go off?
7.5 Conditional Probability
Conditional Probabilities: Probability of an event 𝐴, given some other event B
𝑖𝑠 𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑏𝑦 𝑃(𝐴|𝐵) 𝑎𝑛𝑑 we read it as probability of A given B (within B or under B).
𝑃(𝐴∩𝐵)
𝑛(𝐴∩𝐵)
Formula: P (𝐴|𝐵 ) =
=
(Sample space is now changed to B)
𝑃(𝐵)
𝑛(𝐵)
Product Rule: 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) ∙ 𝑃(𝐵|𝐴) or P (A∩ 𝐵) = 𝑃(𝐵) ∙ 𝑃(𝐴|𝐵)
Ex. 1 A fair die was rolled. What is the probability a 4 is rolled given an even was rolled?
Ex. 2 A purple 6-sided die and a yellow 6-sided die are rolled at the same time (they are both fair
die, numbered 1-6). What is the probability the sum of the dice is greater than 5, if it is known
that the yellow die rolled a number less than 3?
Ex. 3 Two cards are drawn from a deck of 52 one at a time without replacement. What is the
probability that the 2nd card drawn is a diamond, given that the 1st card drawn was also a
diamond?
Ex. 4 Three cards are dealt from a deck of 52 cards without replacement. What is the probability
of the 3rd card being a heart, given the 1st card is a club and the 2nd card is a heart?
Ex. 5 Two cards are drawn from a deck of 52 cards without replacement. What is the probability
that you drew the jack of clubs and a 6 of any suit?
Independent events: Two events are independent if the occurrence of one event doesn’t affect
the probability of the other event. Formally 𝐸, 𝐹 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑖𝑓 P( E | F )  P( E ) or
𝑃(𝐹|𝐸) = 𝑃(𝐹)
Remark: 1. With replacement: Independent , without replacement : Dependent
2. Mutually exclusive: they do not occur simultaneously ( 𝐸 ∩ 𝐹 = ∅)
3. If 𝐸 𝑎𝑛𝑑 𝐹 𝑎𝑟𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑡ℎ𝑒𝑛 𝑃(𝐸 ∩ 𝐹) = 𝑃(𝐸) ∙ 𝑃(𝐹)
Tree Diagram: A way of seeing what is going on (Visual representation): Helpful when
working out probabilities for more than one event (sequential steps, repeated trials, or several
stages of classification)
Ex. 6 A retailer receives a shipment of DVD players from two different, independent companies.
The first shipment, from company A, is known historically to be 8% defective. The second, from
company B, is known to be 15% defective. If one DVD player is selected from each shipment,
what is the probability at least one DVD player is defective?
Ex. 7 In a council election, the percentage of males and females who voted for the two
candidates are shown in the table.
Males
Females
Voted for Colton
29%
18%
Voted for Stacey
31%
22%
a) Find the probability that a randomly selected voter is male, given that the person voted
for Stacey.
b) Find the probability that a randomly selected voter did not vote for Colton.
c) Find the probability that a randomly selected female voter voted for Stacey.
Ex. 8 Use the tree diagram given at right depicting some situation to answer the following
questions. (A, B: mutually exclusive and C, D: mutually exclusive events)
a. 𝑃(𝐶) {Hint: use the fact: (𝐶 ∩ 𝐴) ∪ (𝐶 ∩ 𝐵) = 𝐶 ∩ (𝐴 ∪ 𝐵) = C}
C
𝑃(𝐶|𝐴) = 0.1
A
P(A)=0.35
D
0.15
b. 𝑃(𝐴 ∩ 𝐷)
C
B
D
c. 𝑃(𝐷|𝐵)
d. 𝑃(𝐴 ∪ 𝐶)
Ex. 9 Box A has 3 green, 4 yellow and 2 red marbles in it. Box B has 8 green and 6 yellow
marbles in it. A marble is drawn from Box A and transferred to Box B.
a. Draw a tree diagram representing this experiment.
b. What is the probability green marble is drawn from Box B, given that a green marble was
drawn from Box A?
c. What is the probability a yellow marble is drawn from Box B?
d. What is the probability a red marble is drawn both times?
8.5 Probability Distributions and Expected Value
Random Variable: A function that assigns a real number to each outcome of an
experiment. There are three types of random variables, finite discrete, infinite discrete,
and continuous.
Finite discrete: The random variable can take on a finite number of distinct values. i.e.
Draw five cards from a normal deck of 52 cards without replacement. Count the number
of aces.
Ex. 1 Determine a random variable for the following experiments, then determine the set of all
possible values for the random variable. (Find a random variable 𝑥, and its values)
a) Draw two marbles from a bag containing 10 red, 6 blue, and 2 white marbles, without
replacement.
b) Roll two fair six-sided dice.
1) Determine the number of sixes rolled.
2) Determine the sum of the dice.
Probability Distribution: A table that lists the possible random variables with the
corresponding probabilities.
Histogram: A bar graph that shows a probability distribution or frequency distribution.
Ex. 2 Create the probability distribution for each problem. Then draw the histogram.
a) Roll two fair six-sided dice. Let 𝑟𝑎𝑛𝑑𝑜𝑚 𝑣𝑎𝑟𝑖𝑎𝑙𝑒 𝑥 be the number of sixes rolled.
𝑥=
𝑛𝑢𝑚𝑏𝑒𝑟𝑠 sixes
𝑥1 =0
𝑥2 =1
𝑥2 =2
𝑃(𝑥)
Remark: 𝑃(𝑥𝑖 ) = 𝑃(𝐸𝑖 ), 𝐸𝑖 = {𝑠 ∈ 𝑆| 𝑥(𝑠) = 𝑥𝑖 }
𝑆: Sample space, 𝐸𝑖 : the event whose random
variable value is 𝑥𝑖
b) Roll two fair six-sided dice. Let 𝑥 be the sum of the dice.
𝑥=
𝑠𝑢𝑚 of
the
dice
𝑃(𝑥)
2
3
4
5
6
7
8
9
10
11
12
c). Draw two marbles from a bag containing 10 red, 6 blue, and 2 white marbles, without
replacement. Let 𝑥 𝑏𝑒 the number of red marbles.
𝑥 =# of red
marbles
𝑃(𝑥)
0
1
2
Suppose the random variable 𝑥 can take on the 𝑛values𝑥1 , 𝑥2 , . . . , 𝑥𝑛 . Also, suppose the
probability that these values occur are𝑝1 , 𝑝2 , . . . , 𝑝𝑛 . Then the expected value of the random
variable is 𝐸(𝑥) = 𝑥1 𝑝1 + 𝑥2 𝑝2 + ⋯ + 𝑥𝑛 𝑝𝑛 (the weighted average of values of the random
variable 𝑥)
Ex. 3 Find the expected values of the previous examples.
a)
b)
c)
Games (Expected value = 𝑡ℎ𝑒 Return or winnings you can expect from games)
What makes a player want to play a game? Should the player want to play the game if she has a
better than 50% chance of winning? Take the following game as an example…
Let’s play a game…
Roll one die. If a six is rolled, you win $1. If it isn’t rolled, you lose $1. Do you want to play
this game?
Roll one die. If a six is rolled, you win $1,000. If it isn’t rolled, you lose $1. Now, do you want
to play this game?
The difference between whether you should and should not play a game involves the winnings
AND the probability of winning. The expected value (or expected winnings) of a game will
determine whether you should play a game. If the player has a positive expected winnings, then
he or she SHOULD play, and if it is a negative expected winnings, then he or she SHOULD
NOT play.
Ex. 4 Two players, A and B, play the following game. Player A draws a card from a deck of 52.
If a face card is drawn, Player A wins $80 from Player B. If any other card is drawn, Player A
loses $20 to Player B. What is the expected value for Player A’s winnings? How much should
Player A expect to be up or down after playing the game 10 times?
Ex. 5 Frank pays $1 to draw a ball from a barrel full of possible prizes. In the barrel, there are
the following prizes in the balls, one $500 gift card, four $100 gift cards, and ten $50 gift cards.
If there are a total of 4000 balls in the barrel, find the expected value for Frank’s winnings for the
first drawing.
Definition:
- A fair game has an expected value of 0.
Ex. 6 Sally plays roulette at a casino. There are 38 possible
numbers on the roulette wheel (18 red, 18 black, and 2 green). If
she picks red or black correctly, she wins her money back plus what
she bet. If she picks green correctly, she gets her money back plus
20 times what she bet. She decides to put $100 on red and $50 on
green. What are her expected winnings? Is the game fair? Why or
why not?