Transcript estat4t_0403 - Gordon State College

Lecture Slides
Elementary Statistics
Eleventh Edition
and the Triola Statistics Series
by Mario F. Triola
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4.1 - 1
Chapter 4
Probability
4-1 Review and Preview
4-2 Basic Concepts of Probability
4-3 Addition Rule
4-4 Multiplication Rule: Basics
4-5 Multiplication Rule: Complements and
Conditional Probability
4-6 Counting
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4.1 - 2
Section 4-3
Addition Rule
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4.1 - 3
Key Concept
This section presents the addition rule as a
device for finding probabilities that can be
expressed as P(A or B), the probability that
either event A occurs or event B occurs (or
they both occur) as the single outcome of
the procedure.
The key word in this section is “or.” It is the
inclusive or, which means either one or the
other or both.
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4.1 - 4
Compound Event
Compound Event
any event combining 2 or more simple events
Notation
P(A or B) = P (in a single trial, event A occurs
or event B occurs or they both occur)
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4.1 - 5
General Rule for a
Compound Event
When finding the probability that event
A occurs or event B occurs, find the
total number of ways A can occur and
the number of ways B can occur, but
find that total in such a way that no
outcome is counted more than once.
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4.1 - 6
Compound Event
Formal Addition Rule
P(A or B) = P(A) + P(B) – P(A and B)
where P(A and B) denotes the probability
that A and B both occur at the same time as
an outcome in a trial of a procedure.
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4.1 - 7
Compound Event
Intuitive Addition Rule
To find P(A or B), find the sum of the
number of ways event A can occur and the
number of ways event B can occur, adding
in such a way that every outcome is
counted only once. P(A or B) is equal to
that sum, divided by the total number of
outcomes in the sample space.
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4.1 - 8
Disjoint or Mutually Exclusive
Events A and B are disjoint (or mutually
exclusive) if they cannot occur at the same
time. (That is, disjoint events do not
overlap.)
Venn Diagram for Events That Are
Not Disjoint
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Venn Diagram for Disjoint Events
4.1 - 9
Complementary
Events
P(A) and P(A)
are disjoint
It is impossible for an event and its
complement to occur at the same time.
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4.1 - 10
Rule of
Complementary Events
P(A) + P(A) = 1
P(A) = 1 – P(A)
P(A) = 1 – P(A)
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4.1 - 11
Venn Diagram for the
Complement of Event A
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4.1 - 12
EXAMPLE
The data in the chart below represent the marital
status of males and females 18 years or older in the
US in 1998. Use it to answer the questions on the
next slide.
(Source: US Census
Males
Females
Totals
Bureau)
(in millions) (in millions) (in millions)
Never Married
25.5
21.0
46.5
Married
58.6
59.3
117.9
Widowed
2.6
11.0
13.6
Divorced
8.3
11.1
19.4
Totals
(in millions)
95.0
102.4
197.4 4.1 - 13
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EXAMPLE (CONCLUDED)
1. Determine the probability that a randomly
selected United States resident 18 years or
older is male.
2. Determine the probability that a randomly
selected United States resident 18 years or
older is widowed.
3. Determine the probability that a randomly
selected United States resident 18 years or
older is widowed or divorced.
4. Determine the probability that a randomly
selected United States resident 18 years or
older is male or widowed.
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4.1 - 14
EXAMPLE
The data in the table below represent the income
distribution of households in the US in 2000. (Source:
US Bureau of the Census)
Annual Income
Number (in
thousands)
Annual Income
Number (in
thousands)
Less than $10,000
10,023
$50,000 to $74,999
20,018
$10,000 to $14,999
6,995
$75,000 to $99,999
10,480
$15,000 to $24,999
13,994
$100,000 to $149,999
8,125
$25,000 to $34,999
13,491
$150,000 to $199,999
2,337
$35,000 to $49,999
17,032
$200,000 or more
2,239
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4.1 - 15
EXAMPLE (CONCLUDED)
1. Compute the probability that a randomly
selected household earned $200,000 or more
in 2000.
2. Compute the probability that a randomly
selected household earned less than $200,000
in 2000.
3. Compute the probability that a randomly
selected household earned at least $10,000 in
2000.
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4.1 - 16
Recap
In this section we have discussed:
 Compound events.
 Formal addition rule.
 Intuitive addition rule.
 Disjoint events.
 Complementary events.
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4.1 - 17