Chapter 5
Section 2
The Addition Rule
and Complements
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 1 of 21
Chapter 5 – Section 2
● Learning objectives
1 Use the Addition Rule for disjoint events
2 Use the General Addition Rule
3 Compute the probability of an event using the
Complement Rule
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 2 of 21
Chapter 5 – Section 2
● Learning objectives
1 Use the Addition Rule for disjoint events
2 Use the General Addition Rule
3 Compute the probability of an event using the
Complement Rule
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 3 of 21
Chapter 5 – Section 2
● Venn Diagrams provide a useful way to visualize
probabilities
 The entire rectangle represents the sample space S
 The circle represents an event E
S
E
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 4 of 21
Chapter 5 – Section 2
● In the Venn diagram below




The sample space is {0, 1, 2, 3, …, 9}
The event E is {0, 1, 2}
The event F is {8, 9}
The outcomes {3}, {4}, {5}, {6}, {7} are in neither event
E nor event F
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 5 of 21
Chapter 5 – Section 2
● Two events are disjoint if they do not have any
outcomes in common
● Another name for this is mutually exclusive
● Two events are disjoint if it is impossible for both
to happen at the same time
● E and F below are disjoint
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 6 of 21
Chapter 5 – Section 2
● For disjoint events, the outcomes of (E or F) can
be listed as the outcomes of E followed by the
outcomes of F
● There are no duplicates in this list
● The Addition Rule for disjoint events is
P(E or F) = P(E) + P(F)
● Thus we can find P(E or F) if we know both P(E)
and P(F)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 7 of 21
Chapter 5 – Section 2
● This is also true for more than two disjoint
events
● If E, F, G, … are all disjoint (none of them have
any outcomes in common), then
P(E or F or G or …) = P(E) + P(F) + P(G) + …
● The Venn diagram below is an example of this
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 8 of 21
Chapter 5 – Section 2
● Example
● In rolling a fair die, what is the chance of rolling a
{2 or lower} or a {6}
{6}?
 The probability of {2 or lower} is 2/6
 The probability of {6} is 1/6
 The two events {1, 2} and {6} are disjoint
● The total probability is 2/6 + 1/6 = 3/6 = 1/2
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 9 of 21
Chapter 5 – Section 2
● Learning objectives
1 Use the Addition Rule for disjoint events
2 Use the General Addition Rule
3 Compute the probability of an event using the
Complement Rule
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 10 of 21
Chapter 5 – Section 2
● The addition rule only applies to events that are
disjoint
● If the two events are not disjoint, then this rule
must be modified
● Some outcomes will be double counted
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 11 of 21
Chapter 5 – Section 2
● The Venn diagram below illustrates how the
outcomes {1} and {3} are counted both in event
E and event F
The overlapping
region is (E and F)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 12 of 21
Chapter 5 – Section 2
● Example
● In rolling a fair die, what is the chance of rolling a
{2 or lower} or an even number?





The probability of {2 or lower} is 2/6
The probability of {2, 4, 6} is 3/6
The two events {1, 2} and {2, 4, 6} are not disjoint
The total probability is not 2/6 + 3/6 = 5/6
The total probability is 4/6 because the event is {1, 2,
4, 6}
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 13 of 21
Chapter 5 – Section 2
● For the formula P(E) + P(F), all the outcomes
that are in both events are counted twice
● Thus, to compute P(E or F), these outcomes
must be subtracted (once)
● The General Addition Rule is
P(E or F) = P(E) + P(F) – P(E and F)
● This rule is true both for disjoint events and for
not disjoint events
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 14 of 21
Chapter 5 – Section 2
● Example
● When choosing a card at random out of a deck
of 52 cards, what is the probability of choosing a
queen or a heart?
 E = “choosing a queen”
 F = “choosing a heart”
● E and F are not disjoint (it is possible to choose
the queen of hearts), so we must use the
General Addition Rule
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 15 of 21
Chapter 5 – Section 2
● P(E) = P(queen) = 4/52
● P(F) = P(heart) = 13/52
● P(E and F) = P(queen of hearts) = 1/52, so
P
(queen
or
heart)

P
(queen)

P
(heart)

P
(queen
and
heart)
4 13
1
  
52
52
52
16

52
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 16 of 21
Chapter 5 – Section 2
● Learning objectives
1 Use the Addition Rule for disjoint events
2 Use the General Addition Rule
3 Compute the probability of an event using the
Complement Rule
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 17 of 21
Chapter 5 – Section 2
● The complement of the event E, written Ec,
consists of all the outcomes that are not in that
event
● Examples
 Flipping a coin … E = “heads” … Ec = “tails”
 Rolling a die … E = {even numbers} … Ec = {odd
numbers}
 Weather … E = “will rain” … Ec = “won’t rain”
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 18 of 21
Chapter 5 – Section 2
● The probability of the complement Ec is 1 minus
the probability of E
● This can be shown in one of two ways
 It’s obvious … if there is a 30% chance of rain, then
there is a 70% chance of no rain
 E and Ec are two disjoint events that add up to the
entire sample space
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 19 of 21
Chapter 5 – Section 2
● The Complement Rule can also be illustrated
using a Venn diagram
Entire region
The area of the
region outside the
circle represents Ec
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 20 of 21
Summary: Chapter 5 – Section 2
● Probabilities obey additional rules
● For disjoint events, the Addition Rule is used for
calculating “or” probabilities
● For events that are not disjoint, the Addition Rule
is not valid … instead the General Addition Rule
is used for calculating “or” probabilities
● The Complement Rule is used for calculating
“not” probabilities
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 21 of 21
Examples

According to the U.S. Census Bureau, among
males over the age of 24, 15% did not complete
high school, 31% completed high school, 17%
attended some college without graduating, and
37% are college graduates. (Source: U.S.
Department of Commerce, Census Bureau,
Current Population Survey, March 2004.)
Compute the probability that a randomly selected
male in the United States who is over the age of
24 will not have attended college.
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 22 of 21
Examples
● Let E be the event that a randomly selected person in the U.S. who is
over the age of 24 has attended college. Let F be the event that a
randomly selected U.S. resident over the age of 24 is female.
According to the U.S. Census Bureau, P(E) = 0.47 and P(F) = 0.52,
and P(E and F) is 0.27. (Source: U.S. Department of Commerce,
Census Bureau, Current Population Survey, March 2004.)

a. What is the probability that a randomly selected person in the U.S.
over the age of 24 is a male?

b. What is the probability that a randomly selected person in the U.S.
over the age of 24 is a female or has attended college?

c. What is the probability that a randomly selected person in the U.S.
over the age of 24 is a male and has not attended college?

d. What is the probability that a randomly selected person in the U.S.
over the age of 24 is a female and has not attended college?
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 23 of 21
●
●
●
●
(0.48)
(0.72)
(0.28)
(0.25)
Sullivan – Fundamentals of Statistics – 2nd Edition – Chapter 5 Section 2 – Slide 24 of 21