The Addition Rule

1
Section 4-3

• The Addition Rule

2
COMPOUND EVENT
A compound event is any event combining two or
more simple events.
NOTATION P(A or B) P(event A occurs or event B
occurs or they both occur)
3
GENERAL RULE FOR FINDING THE PROBABILITY OF 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 the total in such a way that no outcome
is counted more than once.
4
FORMAL ADDITION RULE
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.
5
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.
6
DISJOINT EVENTS
Events A and B are disjoint (or mutually
exclusive) if they cannot both occur together.
7
OBSERVATIONS ONDISJOINT EVENTS

• If two events, A and B, are disjoint, then
  P(A and B) 0.
• If events A and B are disjoint, then
• P(A or B) P(A) P(B).

8
APPLYING THE ADDITION RULE
P(A or B) Addition Rule
Are A and B disjoint ?
Yes
P(A or B) P(A) P(B)
Disjoint events cannot happen at the same time.
They are separate, nonoverlapping events.
No
P(A or B) P(A) P(B) - P(A and B)
9
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.
10
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.

11
COMPLEMENTARY EVENTS
Note that events A and are disjoint. Also,
we can be absolutely certain that either A or
occurs. So we have
12
RULE OF COMPLEMENTARY EVENTS
13
VENN DIAGRAM FOR THE COMPLEMENT OF A
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)
15
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.