1. Lecture Slides Elementary Statistics Eleventh Edition
and the Triola Statistics Series
by Mario F. Triola

2. 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

3. Section 4-3
Addition Rule

4. 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.

5. any event combining 2 or more simple events
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)

6. 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.

7. P(A or B) = P(A) + P(B) – P(A and B)
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.

8. 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.

9. 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
Venn Diagram for Disjoint Events

10. Complementary Events P(A) and P(A) are disjoint
It is impossible for an event and its complement to occur at the same time.

11. Rule of Complementary Events
P(A) + P(A) = 1
= 1 – P(A)
P(A) = 1 – P(A)
P(A)

12. Venn Diagram for the Complement of Event A

13. EXAMPLE
The data in the chart below represent the marital status of males and females 18 years or older in the US in Use it to answer the questions on the next slide.
(Source: US Census Bureau)
Males
(in millions)
Females
Totals
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
95.0
102.4
197.4

14. EXAMPLE (CONCLUDED)
Determine the probability that a randomly selected United States resident 18 years or older is male.
Determine the probability that a randomly selected United States resident 18 years or older is widowed.
Determine the probability that a randomly selected United States resident 18 years or older is widowed or divorced.
Determine the probability that a randomly selected United States resident 18 years or older is male or widowed.

15. EXAMPLE
The data in the table below represent the income distribution of households in the US in (Source: US Bureau of the Census)
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

16. EXAMPLE (CONCLUDED)
Compute the probability that a randomly selected household earned $200,000 or more in 2000.
Compute the probability that a randomly selected household earned less than $200,000 in 2000.
Compute the probability that a randomly selected household earned at least $10,000 in 2000.

17. Recap In this section we have discussed: Compound events.
Formal addition rule.
Intuitive addition rule.
Disjoint events.
Complementary events.