The Addition Rules

CHAPTER
3
The Addition Rules
Introduction
In this chapter, the theory of probability is extended by using what are called
the addition rules. Here one is interested in finding the probability of one
event or another event occurring. In these situations, one must consider
whether or not both events have common outcomes. For example, if you are
asked to find the probability that you will get three oranges or three cherries
on a slot machine, you know that these two events cannot occur at the same
time if the machine has only three windows. In another situation you may be
asked to find the probability of getting an odd number or a number less than
500 on a daily three-digit lottery drawing. Here the events have common
outcomes. For example, the number 451 is an odd number and a number less
than 500. The two addition rules will enable you to solve these kinds of
problems as well as many other probability problems.
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Mutually Exclusive Events
Many problems in probability involve finding the probability of two or more
events. For example, when a card is selected at random from a deck, what is
the probability that the card is a king or a queen? In this case, there are two
situations to consider. They are:
1. The card selected is a king
2. The card selected is a queen
Now consider another example. When a card is selected from a deck, find
the probability that the card is a king or a diamond.
In this case, there are three situations to consider:
1. The card is a king
2. The card is a diamond
3. The card is a king and a diamond. That is, the card is the king of

diamonds.
The difference is that in the first example, a card cannot be both a king and
a queen at the same time, whereas in the second example, it is possible for the
card selected to be a king and a diamond at the same time. In the first
example, we say the two events are mutually exclusive. In the second example,
we say the two events are not mutually exclusive. Two events then are
mutually exclusive if they cannot occur at the same time. In other words, the
events have no common outcomes.
EXAMPLE: Which of these events are mutually exclusive?
a. Selecting a card at random from a deck and getting an ace or a club
b. Rolling a die and getting an odd number or a number less than 4
c. Rolling two dice and getting a sum of 7 or 11
d. Selecting a student at random who is full-time or part-time
e. Selecting a student who is a female or a junior
SOLUTION:
a. No. The ace of clubs is an outcome of both events.
b. No. One and three are common outcomes.
c. Yes
d. Yes
e. No. A female student who is a junior is a common outcome.
CHAPTER 3 The Addition Rules
44
Addition Rule I
The probability of two or more events occurring can be determined by using
the addition rules. The first rule is used when the events are mutually
exclusive.
Addition Rule I: When two events are mutually exclusive,
PðA or BÞ¼PðAÞþPðBÞ
EXAMPLE: When a die is rolled, find the probability of getting a 2 or a 3.
SOLUTION:

As shown in Chapter 1, the problem can be solved by looking at the sample
space, which is 1, 2, 3, 4, 5, 6. Since there are 2 favorable outcomes from
6 outcomes, P(2 or 3) ¼
2
6
¼
1
3
. Since the events are mutually exclusive,
addition rule 1 also can be used:
Pð2or3Þ¼Pð2ÞþPð3Þ¼
1
6
þ
1
6
¼
2
6
¼
1
3
EXAMPLE: In a committee meeting, there were 5 freshmen, 6 sophomores,
3 juniors, and 2 seniors. If a student is selected at random to be the
chairperson, find the probability that the chairperson is a sophomore or a
junior.
SOLUTION:
There are 6 sophomores and 3 juniors and a total of 16 students.
P(sophomore or junior) ¼ PðsophomoreÞþPð juniorÞ¼
6

16
þ
3
16
¼
9
16
EXAMPLE: A card is selected at random from a deck. Find the probability
that the card is an ace or a king.
SOLUTION:
P(ace or king) ¼ PðaceÞþPðkingÞ¼
4
52
þ
4
52
¼
8
52
¼
2
13
The word or is the key word, and it means one event occurs or the other
event occurs.
CHAPTER 3 The Addition Rules
45
PRACTICE
1. In a box there are 3 red pens, 5 blue pens, and 2 black pens. If a
person selects a pen at random, find the probability that the pen is
a. A blue or a red pen.

b. A red or a black pen.
2. A small automobile dealer has 4 Buicks, 7 Fords, 3 Chryslers, and
6 Chevrolets. If a car is selected at random, find the probability
that it is
a. A Buick or a Chevrolet.
b. A Chrysler or a Chevrolet.
3. In a model railroader club, 23 members model HO scale, 15 members
model N scale, 10 members model G scale, and 5 members model
O scale. If a member is selected at random, find the probability that
the member models
a. N or G scale.
b. HO or O scale.
4. A package of candy contains 8 red pieces, 6 white pieces, 2 blue
pieces, and 4 green pieces. If a piece is selected at random, find the
probability that it is
a. White or green.
b. Blue or red.
5. On a bookshelf in a classroom there are 6 mathematics books,
5 reading books, 4 science books, and 10 history books. If a student
selects a book at random, find the probability that the book is
a. A history book or a mathematics book.
b. A reading book or a science book.
ANSWERS
1. a. P(blue or red) ¼ P(blue) þ P(red) ¼
5
10
þ
3
10
¼

8
10
¼
4
5
b. P(red or black) ¼ P(red) þ P(black) ¼
3
10
þ
2
10
¼
5
10
¼
1
2
CHAPTER 3 The Addition Rules
46
2. a. P(Buick or Chevrolet) ¼ P(Buick) þ P(Chevrolet)
¼
4
20
þ
6
20
¼
10
20
¼

1
2
b. P(Chrysler or Chevrolet) ¼ P(Chrysler) þ P(Chevrolet)
¼
3
20
þ
6
20
¼
9
20
3. a. P(N or G) ¼ P(N) þ P(G) ¼
15
53
þ
10
53
¼
25
53
b. P(HO or O) ¼ P(HO) þ P(O) ¼
23
53
þ
5
53
¼
28
53

4. a. P(white or green) ¼ P(white) þ P(green) ¼
6
20
þ
4
20
¼
10
20
¼
1
2
b. P(blue or red) ¼ P(blue) þ P(red) ¼
2
20
þ
8
20
¼
10
20
¼
1
2
5. a. P(history or math) ¼ P(history) þ P(math) ¼
10
25
þ
6
25

¼
16
25
b. P(reading or science) ¼ P(reading) þ P(science) ¼
5
25
þ
4
25
¼
9
25
Addition Rule II
When two events are not mutually exclusive, you need to add the probabilities
of each of the two events and subtract the probability of the outcomes that
are common to both events. In this case, addition rule II can be used.
Addition Rule II: If A and B are two events that are not mutually exclusive,
then PðA or BÞ¼PðAÞþPðBÞÀPðA and BÞ, where A and B means the num-
ber of outcomes that event A and event B have in common.
CHAPTER 3 The Addition Rules
47