Math 141
Chapter 7 : Probability
TAMU - Summer 2014
1.3 Sample Space and Events
Experiment: is an activity that has observable results. The result of an experiment is knows as
its outcome.
Standard deck of 52 playing cards: A standard deck of 52 playing cards has four 13-card suits:
clubs, diamonds, hearts, and spades. The diamonds and hearts are red, the clubs and spades are
black. Each 13-card suit contains cards numbered from 2 to 10, a jack(11), a queen(12), a
king(13), and an ace(14).
Sample Space ( S ): of an experiment is the set of all possible outcomes of the experiment
(similar to the Universal Set: set of all possible elements). Each repetition of an experiment is
known as a Trial.
Examples:
Experiment
Flip a coin
Roll a 6-side fair die
Select a ball from a basket having 2 Red and 3
Green balls
Flip two coins
Flip a coin twice
Roll two fair 6-side dice and the sum of the
numbers on the dice is 8.
Roll two fair 4side dice and the product of the
numbers on the dice is at most 5.
Flip a coin three times in a row
Roll a die, and if the number on the die is even,
then flip a coin.
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Sample Space
Math 141
Chapter 7 : Probability
TAMU - Summer 2014
Event: Given a sample space S for an experiment, an Event is any subset of S. An elementary
(or simple) event is an event with a single outcome.
e.g. Flip two coins and list all the elementary events.

Since Æ is a subset of every set, Æ Í S. Æ is the “impossible event” since it means no
outcome has occurred.
e.g.: Flipping a coin, the event that both heads and tails show is an impossible event.
H È T = Æ.

Every set is a subset of itself. Hence, S Í S . S is the “certainty event”, since any
outcome of an experiment must be in S.
Question
Flip two coins and find the event that at least
one tail comes up.
Flip a coin and find the event that both are
heads.
Roll two dice and find the event that the
numbers of top faces of both dice are even.
Roll two dice and find the event that the sum of
numbers on top faces of both dice is less than
5.
Pick a card from a stack of 52-cards and
observe the suit.
A card is picked from the standard deck and
the color is observed.
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Event
Math 141
Chapter 7 : Probability
TAMU - Summer 2014
Example: Two fair 4-faced dice are rolled and the numbers on each are observed. List all the
elements in the sample space of this experiment.
a) What is the sample space?
Alternative way: to find the total number of elements in Sample Space:
b) How many elements (outcomes) are there in the sample space?
c) How many events are possible for this sample space?
d) What is the sample space for “sum of the dots on the top faces of both the dice?”
e) What is the sample space for “ the product of the dots on the top faces of both the dice “
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Math 141
Chapter 7 : Probability
TAMU - Summer 2014
Uniform Sample Space : is one where each event is equally likely.
Example: An experiment consists of selecting a digit from the number 112964333 and observing
it.
a) What is the uniform sample space for this experiment?
b) Indicate the outcomes in the event that “an even digit” is selected.
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Math 141
Chapter 7 : Probability
TAMU - Summer 2014
Union, Intersection, and Complement of Events
If E and F are two events of an experiment, with sample space S, where
S = { a, b, c, d, e, f, g, h }
E = { a, b, c, d }
F = { c, a, f }
a) E Ç F: is the union of E and F , and consists of the set of outcomes that are in E or in F
(or both). If an element exists in both the sets, it will be written only once in set E Ç F .
b) E È F : is the intersection of E and F, and consists of the set of outcomes that are in both
E and F .

If E È F =Æ , E and F are said to be “mutually exclusive” events. That is, the sets E and
are F disjoint.
c) EC : is the complement of , E and consists of the set of outcomes not in E. In other words,
E' is the event that “ E does not occur.”
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Math 141
Chapter 7 : Probability
TAMU - Summer 2014
Example : Let a card be chosen from a standard deck of 52 cards. Let E be the event of
drawing a 5. Let F be the event of drawing a heart. Let G be the event of drawing a black.
a) List the elements of E, F.
b) What is E È F , G È F , E È G.
c) Are any of these events mutually exclusive?
d) How many elements are there in E È GC ? E Ç GC ?
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Math 141
Chapter 7 : Probability
TAMU - Summer 2014
Probability
1. A basket contains 3 red and 5 black balls. One ball is chosen at random. What is the
probability that the ball chosen is
a) Black
b) Red
c) Black and Red
d) Black or Red
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Math 141
Chapter 7 : Probability
TAMU - Summer 2014
2. Roll two 6-sided fair dice and observe the sum of the numbers on the top faces of the dice.
What is the probability that the sum is
a) between and including 3 and 6 ?
b) less than 8?
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Math 141
Chapter 7 : Probability
TAMU - Summer 2014
3. A single card is randomly drawn from a standard deck of cards. What is the probability that the
card is either red or a king?
4. A family has three children. Assuming a boy is as likely as a girl to have been born, what is the
probability that 2 are boys and 1 is a girl?
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Math 141
Chapter 7 : Probability
TAMU - Summer 2014
5. A manufactured item is guaranteed for one year and has three critical parts. It has been decided
that during the first year, the probability of failure of the first part is 0.03, of the second part is
0.02, of the third part is 0.01, both the first and the second is 0.005, both the first and the third is
0.003, and all three parts is 0.001. what Is the probability that
a) Exactly one of these parts will fail in the first year?
b) At least one of these three parts fail in the first year?
c) None of these parts fail in the first year?
d) Exactly two of these parts fail in the first year?
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Math 141
Chapter 7 : Probability
TAMU - Summer 2014
6. Two 4-sided dice are rolled. Find the probability distribution table for the sum of the numbers
observed on the top faces.
7. An unfair 6-sided die is rolled 1000 times and the number on top face is observed. The table
shows the results. Find the empirical probability that (a) a 2 will occur (b) an odd number will
occur.
Outcome
1
2
3
4
5
6
Frequency
161
179
148
177
210
125
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Math 141
Chapter 7 : Probability
TAMU - Summer 2014
8. Given P(E) = 0.26, P( F ) = 0.28, P(E È F) = 0.08, what are the odds that
a) E occurs
b) only E occurs
c) E does not occur
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Math 141
Chapter 7 : Probability
9. If odds in favor of E are 13/37, then find
a) P( E )
b) P( EC )
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TAMU - Summer 2014
Math 141
Chapter 7 : Probability
TAMU - Summer 2014
Binomial Trials:
a) binompdf – Exactly k successes
2nd VARS → scroll down to A : binompdf( → Enter n → Press the Comma key → Enter p →
Press the Comma key → Enter k → ENTER
where n : Total number of trials ; p : probability of a success ; k : number of successes
b) binomcdf – at most k successes
2nd VARS → scroll down to B : binomcdf( → Enter n → Press the Comma key → Enter p →
Press the Comma key → Enter k → ENTER
Note : “binomcdf” function is also used to calculate “at least” k successes.
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Math 141
Chapter 7 : Probability
TAMU - Summer 2014
1. An unfair coin is flipped 8 times in succession. The probability of getting a heads is 1/3
a) How many elements are there in the sample space ?
b) What is the probability of getting at least two heads occur?
3. An unfair coin is tossed 100 times in succession. The probability of getting a heads on a toss
is 1/3. What is the probability of getting
a) exactly 40 heads
b) At least 40 heads
c) At most 40 heads
d) More than 40 but less than 60 heads
e) exactly 40 tails
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Math 141
Chapter 7 : Probability
TAMU - Summer 2014
4. A machine produces defective parts 20% of the time. What is the probability that of the next 7
parts,
a) Exactly 3 are defective
b) At most 3 are defective
c) At least 3 are defective
5. Your quiz has 10 multiple-choice questions over the material that has not been covered in
class. You guess on each question. What is the probability of you getting exactly 7 questions
correct?
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