Math 2
Lesson 7-3: The Addition Counting Principle,
Mutually Exclusive, and Complements
Name _____________________________
Date ___________________________
Learning Goals:
 I can apply the Addition Rule to determine and interpret the probability of the union of two events
using the formula P(A or B) = P(A) + P(B) – P(A and B).
 I can explain and apply the meaning of mutually exclusive events and complementary events.
Example 1:
A die is rolled. What is the probability that the number is even and less than 4?
Practice #1
1. A die is rolled. What is the probability that the number is prime and less than 5?
2. A set of polygons contains a square, a rectangle, a rhombus and a trapezoid. If one polygon is chosen at
random, what is the probability that the polygon has all sides equal in length and all right angles?
(Hint: A square and a rhombus have all sides equal in length. A square and a rectangle have all right
angles.)
3. Two dice are rolled, what is the probability of throwing a 4 on the first die and a 3 on the second die?
 Two events that have NO outcomes in common are called mutually exclusive. These are events
that cannot occur at the same time. If two events are mutually exclusive, then P(A and B) = 0
Example 2
A pair of dice is rolled. The events of rolling a sum of 6 and of rolling a double have the outcome (3,3) in
common. These two events are NOT mutually exclusive.
Example 3
A pair of dice is rolled. The events of rolling a sum of 9 and rolling a doubles have
NO outcomes in common. These two events ARE mutually exclusive.
OVER 
Page 2
The rule for “OR” takes into account those values that may get counted more than once when the probability is
determined.
Another way to state this is:
The probability of the union of two events is given by:
P(A∪B) = P(A) + P(B) – P(A∩B)
 Here, P(A) is the probability of event A, P(B) is the probability of event B.
 Also, P(A∩B) is the probability of the intersection of events A and B.
Example 4:
A die is rolled. What is the probability that the number is even or less than 4?
Solution:
(rolling a 2 is in both events)
Answer: Probability = P(A) + P(B) − P(A and B)
= 3/6 + 3/6 − 1/6
= 5/6
Practice #2
1. A die is rolled. What is the probability that the number is prime or less than 5?
Page 3
2. A set of polygons contains a square, a rectangle, a rhombus and a trapezoid. If one polygon is chosen at random,
what is the probability that the polygon has all sides equal in length or all right angles? (Hint: A square
and a
rhombus have all sides equal in length. A square and a rectangle have all right angles.)
3. Two dice are rolled, what is the probability of throwing a 4 on the first die or a 3 on the second die?
4. A piggybank contains 2 quarters, 3 dimes, 4 nickels, and 5 pennies. One coin is removed at random. What is the
probability that the coin is a dime or a nickel?
If A is an event within the sample space S of an activity or experiment, the complement of A(denoted A') consists of
all outcomes in S that are not in A.
The complement of A is everything else in the problem that is NOT in A.
Consider these experiments where an event and its complement are shown:
Experiment: Tossing a coin
Event
A
The coin shows heads.
Complement
A'
The coin shows tails.
Experiment: Drawing a card
Event
A
The card is black.
Complement
A'
The card is red.
The probability of the complement of an event is one minus the probability of the event.
Complement:
P  A '   1  P  A
Example 5:
Solution:
Use the complement to find the probability of rolling two dice and getting a sum of less than 11.
OVER 
Page 4
Practice #3
1. One die is rolled. Two possible events are rolling a number less than 5
and rolling a number which is a multiple of 5. Are these two events
mutually exclusive? Explain
2. A pair of dice is rolled. Two possible events are rolling a sum that is
greater than 8 and rolling a sum that is an even number. Are these two
events mutually exclusive events? Explain.
3. A pair of dice is rolled. A possible event is rolling a sum that is a multiple of 5. What is the complement of this
event? Find the probability of the complement of this event.
4. A pair of dice is rolled. Two possible events are rolling a number which is a multiple of 3 and rolling a number
which is a multiple of 5. Are these two events mutually exclusive? Explain.
5. A pair of dice is rolled and the resulting sum is odd. What is the complement of this event? What is the
probability of the complement?
6. A bag contains marbles of three different colors: 8 black, 6 white, and 4 red. Three marbles are selected at
random, without replacement. Find the probability that the selection contains each of the outcomes listed
below. Express the answer as a decimal to the nearest hundredth.
a.) three black marbles
b.) three white marbles
c.) one white marble followed by two black marbles
d.) a red, a black and a white marble in that order
Math 2
Homework: Lesson 7-3
Name ______________________________
Date _____________________________
1. A paper bag contains 15 slips of paper. Eight of them are blue and are numbered from 1 to 8. Seven of
them are pink and are numbered from 1 to 7. A slip is drawn from the bag and then
replaced. A
second slip is drawn. Are these two events independent?
2. A paper bag contains 15 slips of paper. Eight of them are blue and are numbered from 1 to 8. Seven of
them are pink and are numbered from 1 to 7. A possible event is that a blue slip is chosen. What is
the
probability of the complement of this event?
3. A paper bag contains 15 slips of paper. Eight of them are blue and are numbered from 1 to 8. Seven of
them are pink and are numbered from 1 to 7. You wish to draw a blue slip on the first draw and
a pink slip on the second draw. The first slip is NOT replaced after being drawn. Are these events
independent?
4.
Which of the following pairs of events are mutually exclusive? Justify your answer.

You roll a sum of 7 with a pair of dice; you get doubles on the same roll.

You roll a sum of 8 with a pair of dice; you get doubles on the same roll.

Isaac wears tennis shoes today to math class; Isaac wears dress shoes today to math class.

Sarah owns white shoes; Sarah owns black shoes.
OVER 
5. The table below gives the percentage of high school sophomores who say they engage in various
activities at least once a week.
Weekly Activites of High School Sophomores
Activity
Percentage of Sophomores
Use personal computer at home
71.2
Drive or ride around
56.7
Work on hobbies
41.8
Take sports lesson
22.6
Take class in music, art, language
19.5
Perform community service
10.6
Use the data in the table to help answer, if possible, each of the following questions. If a question
cannot be answered, explain why not.

What is the probability that a randomly selected sophomore takes sports lessons at least once a
week?

What is the probability that a randomly selected sophomore works on hobbies at least once a week?

What is the probability that a randomly selected sophomore takes sports lessons at least once a week
or works on hobbies at least once a week?

What is the probability that a randomly selected sophomore works on hobbies at least once a week
or uses a personal computer at home at least once a week?
6. Suppose two events A and B are mutually exclusive. Which of the Venn diagrams below best represents
this situation? Explain.
I.
II.