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ELEMENTARY STATISTICS 3E

William Navidi and Barry Monk

©McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.

Basic Concepts in Probability

Section 5.1

©McGraw-Hill Education.

Objectives

1.

2.

3.

4.

Construct sample spaces

Compute and interpret probabilities

Approximate probabilities using the Empirical Method

Approximate probabilities by using simulation

©McGraw-Hill Education.

Objective 1

Construct sample spaces

©McGraw-Hill Education.

Probability

A probability experiment is one in which we do not know what any

individual outcome will be, but we do know how a long series of

repetitions will come out.

For example, if we toss a fair coin, we do not know what the

outcome of a single toss will be, but we do know what the outcome

of a long series of tosses will be – about half “heads” and half

“tails”.

The probability of an event is the proportion of times that the event

occurs in the long run. So, for a “fair” coin, that is, one that is

equally likely to come up heads as tails, the probability of heads is

1/2 and the probability of tails is 1/2.

©McGraw-Hill Education.

Law of Large Numbers

The Law of Large Numbers says that as a probability

experiment is repeated again and again, the proportion of

times that a given event occurs will approach its

probability.

©McGraw-Hill Education.

Sample Space

The collection of all the possible outcomes of a probability

experiment is called a sample space.

Example:

Describe the sample space for each of the following:

a) The toss of a coin

The sample space is {Heads, Tails}.

b) The roll of a die

The sample space is {1, 2, 3, 4, 5, 6}

c) The selection of a student at random from a list of 10,000

at a large university

The sample space consists of the 10,000 students.

©McGraw-Hill Education.

Probability Model

We are often concerned with occurrences that consist of several

outcomes. For example, when rolling a die, we might be interested

in the probability of rolling an odd number. Rolling an odd number

corresponds to the collection of outcomes {1, 3, 5} from the sample

space {1, 2, 3, 4, 5, 6}. In general, a collection of outcomes of a

sample space is called an event.

Once we have a sample space, we need to specify a probability of

each event. This is done with a probability model. We use the letter

“ ” to denote probabilities. For example, we denote the probability

that a tossed coin lands heads by (Heads).

In general, if A denotes an event, the probability of event A is

denoted by (A).

©McGraw-Hill Education.

Objective 2

Compute and interpret probabilities

©McGraw-Hill Education.

Probability Rules

The probability of an event is always between 0 and 1. In

other words, for any event A, 0 ≤ (A) ≤ 1.

• If A cannot occur, then (A) = 0.

• If A is certain to occur, then (A) = 1.

©McGraw-Hill Education.

Probabilities with Equally Likely Outcomes

If a sample space has equally likely outcomes and an event A has

Number of outcomes in A

outcomes, then (A) =

=

Number of outcomes in the sample space

Example:

In the Georgia Cash-4 Lottery, a winning number between 0000 and

9999 is chosen at random, with all the possible numbers being equally

likely. What is the probability that all four digits are the same?

The outcomes in the sample space are the numbers from 0000 to

9999, so there are 10,000 equally likely outcomes in the sample space.

There are 10 outcomes for which all the digits are the same: 0000,

1111, 2222, and so on up to 9999.

(All four digits the same) =

©McGraw-Hill Education.

10

10,000

= 0.001

Sampling is a Probability Experiment

Sampling an individual from a population is a probability

experiment. The population is the sample space and

members of the population are equally likely outcomes.

©McGraw-Hill Education.

Example: Sampling as a Probability Experiment

There 10,000 families in a certain town categorized as follows:

Own a house

Own a condo

Rent a house

Rent an apartment

4753

1478

912

2857

A pollster samples a single family from this population.

a) What is the probability that the sampled family owns a house?

b) What is the probability that the sampled family rents?

Solution:

a) The sample space consists of 10,000 households. Of these, 4753

4753

own a house. Therefore, (Own a house) =

= 0.4753.

10,000

b) The number of families who rent is 912 + 2857 = 3769. Therefore,

3769

(Rents) =

= 0.3769.

10,000

©McGraw-Hill Education.

Unusual Events

An unusual event is one that is not likely to happen. In other words, an

event whose probability is small. There are no hard-and-fast rules as to

just how small a probability needs to be before an event is considered

unusual, but we will use the following rule of thumb.

Any event whose probability is less than 0.05 is considered to be

unusual.

Example:

In a college of 5000 students, 150 are math majors. A student is

selected at random and turns out to be a math major. Is this unusual?

Solution:

The event of choosing a math major consists of 150 students out of a

150

total of 5000. The probability of choosing a math major is

= 0.03.

5000

This would be considered an unusual event.

©McGraw-Hill Education.

Objective 3

Approximate probabilities using the Empirical Method

©McGraw-Hill Education.

Approximating Probabilities – Empirical Method

The Law of Large Numbers says that if we repeat a

probability experiment a large number of times, then the

proportion of times that a particular outcome occurs is likely

to be close to the true probability of the outcome.

The Empirical Method consists of repeating an experiment a

large number of times, and using the proportion of times an

outcome occurs to approximate the probability of the

outcome.

©McGraw-Hill Education.

Example: Empirical Method

In a recent year, there were 2,046,935 boys and 1,952,451 girls

born in the U.S. Approximate the probability that a newborn baby

is a boy.

Solution:

We compute the number of times the experiment has been

repeated:

2,046,935 + 1,952,451 = 3,999,386 births.

The proportion of births that are boys is

approximate (Boy) ≈ 0.5118.

©McGraw-Hill Education.

2,046,935

3,999,386

= 0.5118. We

Objective 4

Approximate probabilities by using simulation

©McGraw-Hill Education.

Simulation

In practice, it can be difficult or impossible to repeat an

experiment many times in order to approximate a

probability with the Empirical Method. In some cases, we

can use technology to repeat an equivalent virtual

experiment many times. Conducting a virtual experiment in

this way is called simulation.

©McGraw-Hill Education.

Simulation on the TI-84 PLUS

The randInt command on the TI-84 PLUS calculator may

be used to simulate the rolling of a single die. To access

this command, press MATH, scroll to the PRB menu, and

select randInt. The following screenshots illustrate how

to simulate the rolling of a die 100 times. The outcomes

are stored in list L1.

©McGraw-Hill Education.

Simulation in EXCEL

Suppose that three dice are rolled, the smallest possible total is 3 and the largest possible

total is 18. Technology, such as EXCEL, can be used to “roll” the dice as many times as

desired. Following are the frequencies for each possible outcome when the dice are “rolled”

1000 times. We may, for example, estimate the probability of obtaining the number 10 as

128

(10) = 1000 = 0.128.

©McGraw-Hill Education.

You Should Know . . .

• The Law of Large Numbers

• The definition of:

– Probability

– Sample space

– Event

• How to compute probabilities with equally likely

outcomes

• The rule-of-thumb for determining when an event A is

unusual (if (A) < 0.05)
• How to approximate probabilities using the Empirical
Method
©McGraw-Hill Education.
ELEMENTARY STATISTICS 3E
William Navidi and Barry Monk
©McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
The Addition Rule and
the Rule of Complements
Section 5.2
©McGraw-Hill Education.
Objectives
1. Compute probabilities by using the General Addition Rule
2. Compute probabilities by using the Addition Rule for
Mutually Exclusive Events
3. Compute probabilities by using the Rule of Complements
©McGraw-Hill Education.
Objective 1
Compute probabilities by using the General Addition
Rule
©McGraw-Hill Education.
A or B Events and the General Addition Rule
A compound event is an event that is formed by
combining two or more events. One type of compound
event is of the form A or B. The event A or B occurs
whenever A occurs, B occurs, or A and B both occur.
Probabilities of events in the form A or B are computed
using the General Addition Rule.
General Addition Rule:
For any two events A and B,
(A or B) = (A) + (B) – (A and B)
©McGraw-Hill Education.
Example: General Addition Rule
1000 adults were asked whether they favored a law that would
provide support for higher education. In addition, each person was
classified as likely to vote or not likely to vote based on whether they
voted in the last election. What is the probability that a randomly
selected adult is likely to vote or favors the law?
©McGraw-Hill Education.
Example: General Addition Rule (Solution)
What is the probability that a randomly selected adult is likely to vote or
favors the law?
There are 372 + 262 + 87 = 721 people who are likely to vote, so
(Likely vote) = 721/1000 = 0.721. There are 372 + 151 = 523 people who
favor the law, so (Favor) = 523/1000 = 0.523.
The number of people who are both likely to vote and who favor the law is
372. So, (Likely vote AND Favors) = 372/1000 = 0.372.
By the General Addition Rule,
(Likely vote or Favors) = (Likely vote)+ (Favors)– (Likely vote and favors)
= 0.721 + 0.523 – 0.372
= 0.872
©McGraw-Hill Education.
Objective 2
Compute probabilities by using the Addition Rule for
Mutually Exclusive Events
©McGraw-Hill Education.
Mutually Exclusive Events
Two events are said to be mutually exclusive if it is impossible for
both events to occur.
Example:
A die is rolled. Event A is that the die comes up 3, and event B is that
the die comes up an even number.
These events are mutually exclusive since the die cannot both come
up 3 and come up an even number.
A fair coin is tossed twice. Event A is that one of the tosses is heads,
and Event B is that one of the tosses is tails.
These events are not mutually exclusive since, if the two tosses are
HT or TH, then both events occur.
©McGraw-Hill Education.
The Addition Rule for Mutually Exclusive Events
If events A and B are mutually exclusive, then
(A and B) = 0.
This leads to a simplification of the General Addition Rule.
Addition Rule for Mutually Exclusive Events:
If A and B are mutually exclusive events, then
(A or B) = (A) + (B)
©McGraw-Hill Education.
Example: Addition Rule/Mutually Exclusive
In a recent year of the Olympic Games, a total of 11,544 athletes
participated. Of these, 554 represented the United States, 314
represented Canada, and 125 represented Mexico. What is the
probability that an Olympic athlete chosen at random represents the
U.S. or Canada?
Solution:
These events are mutually exclusive, because it is impossible to
compete for both the U.S. and Canada. So,
(U.S. or Canada) = (U.S.) + (Canada)
554
314
=
+
11,544
11,544
868
=
11,544
= 0.075191
©McGraw-Hill Education.
Objective 3
Compute probabilities by using the Rule of
Complements
©McGraw-Hill Education.
The Complement of an Event
If there is a 60% chance of rain today, then there is a 40%
chance that it will not rain. The events “Rain” and “No rain”
are complements. The complement of an event says that
the event does not occur.
If A is any event, the complement of A is the event that A
does not occur. The complement of A is denoted Ac.
©McGraw-Hill Education.
Example: Complement of an Event
Two hundred students were enrolled in a Statistics class.
Find the complements of the following events:
• Exactly 50 of them are business majors.
The complement is that the number of business majors
is not 50.
• More than 50 of them are business majors.
The complement is that 50 or fewer are business
majors.
• At least 50 of them are business majors.
The complement is that fewer than 50 are business
majors.
©McGraw-Hill Education.
The Rule of Complements
The Rule of Complements:
(Ac) = 1 – (A)
Example:
According to the Wall Street Journal, 40% of cars sold in a
recent year were small cars. What is the probability that a
randomly chosen car sold in that year is not a small car?
Solution:
(Not a small car) = 1 – (Small car) = 1 – 0.40 = 0.60.
©McGraw-Hill Education.
You Should Know . . .
• How to use The General Addition Rule to compute
probabilities of events in the form A or B
• How to determine whether events are mutually exclusive
• How to compute probabilities of mutually exclusive
events
• How to determine the complement of an event
• How to use the Rule of Complements to compute
probabilities
©McGraw-Hill Education.
ELEMENTARY STATISTICS 3E
William Navidi and Barry Monk
©McGraw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of McGraw-Hill Education.
Random Variables
Section 6.1
©McGraw-Hill Education.
Objectives
1. Distinguish between discrete and continuous random
variables
2. Determine a probability distribution for a discrete random
variable
3. Describe the connection between probability distributions
and populations
4. Construct a probability histogram for a discrete random
variable
5. Compute the mean of a discrete random variable
6. Compute the variance and standard deviation of a discrete
random variable
©McGraw-Hill Education.
Objective 1
Distinguish between discrete and continuous random
variables
©McGraw-Hill Education.
Random Variable
If we roll a fair die, the possible outcomes are the numbers 1, 2, 3, 4,
5, and 6, and each of these numbers has probability 1/6. Rolling a
die is a probability experiment whose outcomes are numbers. The
outcome of such an experiment is called a random variable.
A random variable is a numerical outcome of a probability
experiment.
©McGraw-Hill Education.
Discrete and Continuous Random Variables
Discrete random variables are random variables whose
possible values can be listed. Examples include:
• The number that comes up on the roll of a die.
• The number of siblings a randomly chosen person
has.
Continuous random variables are random variables that
can take on any value in an interval. Examples include:
• The height of a randomly chosen college student.
• The amount of electricity used to light a randomly
chosen classroom.
©McGraw-Hill Education.
Objective 2
Determine a probability distribution for a discrete
random variable
©McGraw-Hill Education.
Probability Distribution
A probability distribution for a discrete random variable
specifies the probability for each possible value of the
random variable.
Properties:
• 0 ≤ ≤ 1 for every possible
• ∑ = 1
©McGraw-Hill Education.
Example 1: Probability Distribution
Decide if the following represents a probability distribution.
x
1
2
P(x)
0.25
0.65
3
4
–0.30
0.11
This is not a probability distribution. (3) is not between 0 and 1.
©McGraw-Hill Education.
Example 2: Probability Distribution
Decide if the following represents a probability distribution.
x
–1
–0.5
P(x)
0.17
0.25
0
0.5
1
0.31
0.22
0.05
This is a probability distribution. All the probabilities are between 0
and 1, and they add up to 1.
©McGraw-Hill Education.
Example 3: Probability Distribution
Decide if the following represents a probability distribution.
x
1
10
100
1000
P(x)
1.02
0.31
0.90
0.43
This is not a probability distribution. (1) is not between 0 and 1.
©McGraw-Hill Education.
Example: Computing Probabilities (Part a)
Four patients have made appointments to have
their blood pressure checked at a clinic. Let be
the number of them that have high blood
pressure. The probability distribution of is as
follows.
x
0
P(x)
0.23
1
2
3
0.41
0.27
0.08
a) Find (2 or 3)
4
0.01
Solution:
The events “2” and “3” are mutually exclusive, since they cannot
both happen. We use the Addition Rule for Mutually Exclusive
events:
(2 or 3) = (2) + (3) = 0.27 + 0.08 = 0.35
©McGraw-Hill Education.
Example: Computing Probabilities (Part b)
Four patients have made appointments to have
their blood pressure checked at a clinic. Let be
the number of them that have high blood
pressure. The probability distribution of is as
follows.
x
0
P(x)
0.23
1
2
3
0.41
0.27
0.08
b) Find (More than 1)
4
0.01
Solution:
“More than 1” means “2 or 3 or 4.” We use the Addition Rule for
Mutually Exclusive events:
(More than 1) = (2 or 3 or 4) = 0.27 + 0.08 + 0.01 = 0.36
©McGraw-Hill Education.
Example: Computing Probabilities (Part c)
Four patients have made appointments to have
their blood pressure checked at a clinic. Let be
the number of them that have high blood
pressure. The probability distribution of is as
follows.
x
0
P(x)
0.23
1
2
3
0.41
0.27
0.08
c) Find (At least 1)
4
0.01
Solution:
We use the Rule of Complements. Recall that the complement of “At
least one” is “none”:
(At least one) = 1 – (0) = 1 – 0.23 = 0.77
©McGraw-Hill Education.
Objective 3
Describe the connection between probability
distributions and populations
©McGraw-Hill Education.
Probability Distributions and Populations
Statisticians are interested in studying samples drawn from
populations. Random variables are important because when
an item is drawn from a population, the value observed is
the value of a random variable.
The probability distribution of the random variable tells how
frequently we can expect each of the possible values of the
random variable to turn up in the sample.
©McGraw-Hill Education.
Example: Connection with Populations
An airport parking facility contains 1000 parking spaces. Of these,
142 are covered long-term spaces that cost $2.00 per hour, 378
are covered short-term spaces that cost $4.50 per hour, 423 are

uncovered long-term spaces that cost $1.50 per hour, and 57 are

uncovered short-term spaces that cost $4.00 per hour. A parking

space is selected at random. Let represent the hourly parking

fee for the randomly sampled space. Find the probability

distribution of .

Solution:

To find the probability distribution, we must list the possible

values of and then find the probability of each of them. The

possible values of are 1.50, 2.00, 4.00, 4.50. Next, we find their

probabilities.

©McGraw-Hill Education.

Example: Connection with Populations (Continued)

An airport parking facility contains 1000 parking spaces. Of these, 142 are

covered long-term spaces that cost $2.00 per hour, 378 are covered shortterm spaces that cost $4.50 per hour, 423 are uncovered long-term spaces

that cost $1.50 per hour, and 57 are uncovered short-term spaces that cost

$4.00 per hour.

1.50

2.00

4.00

4.50

©McGraw-Hill Education.

# of spaces costing $1.50

423

=

=

= 0.423

total # of spaces

1000

# of spaces costing $2.00

142

=

=

= 0.142

total #of spaces

1000

# of spaces costing $4.00

57

=

=

= 0.057

total # of spaces

1000

# of spaces costing $4.50

378

=

=

= 0.378

total # of spaces

1000

x

1.50

2.00

4.00

4.50

P(x)

0.423

0.142

0.057

0.378

ҧ is a Random Variable

Often when we draw a s …

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