
Addition Rules
for Probability 

Unit 6 > Lesson 6 of 12 
Experiment 1: 
A single 6sided die is rolled. What is the probability of rolling a 2 or a 5? 

Possibilities: 
1. 
The number rolled can be a 2. 
2. 
The number rolled can be a 5. 

Events: 
These events are mutually exclusive since they cannot occur at the same time. 
Probabilities: 
How do we find the probabilities of these mutually exclusive events? We need a rule to
guide us.

Addition Rule 1: 
When two events, A and B, are mutually exclusive, the probability that A or B will occur
is the sum of the probability of each event. 

P(A or B) = P(A) + P(B) 
Let's use this addition rule to find the probability for Experiment 1.
Experiment 1: 
A single 6sided die is rolled. What is the probability of rolling a 2 or a 5? 

Probabilities: 
P(2) 
= 
1 
6 

P(5) 
= 
1 
6 

P(2 or 5) 
= 
P(2) 
+ 
P(5) 


= 
1 
+ 
1 
6 
6 

= 
2 
6 

= 
1 
3 

Experiment 2: 
A spinner has 4 equal sectors colored yellow, blue, green, and red. What is the
probability of landing on red or blue after spinning this spinner?



Probabilities: 
P(red) 
= 
1 
4 

P(blue) 
= 
1 
4 

P(red or blue) 
= 
P(red) 
+ 
P(blue) 


= 
1 
+ 
1 
4 
4 

= 
2 
4 

= 
1 
2 

Experiment 3: 
A glass jar contains 1 red, 3 green, 2 blue, and 4 yellow marbles. If a
single marble is chosen at random from the jar, what is the probability that it is yellow or green?



Probabilities: 
P(yellow) 
= 
4 
10 

P(green) 
= 
3 
10 

P(yellow or green) 
= 
P(yellow) 
+ 
P(green) 


= 
4 
+ 
3 
10 
10 

= 
7 
10 

In each of the three experiments above, the events are mutually exclusive. Let's look
at some experiments in which the events are nonmutually exclusive. 
Experiment 4: 
A single card is chosen at random from a standard deck of 52 playing
cards. What is the probability of choosing a king or a club?



Probabilities: 
P(king or club) 
= 
P(king) 
+ 
P(club) 
 
P(king of clubs) 


= 
4 
+ 
13 
 
1 
52 
52 
52 

= 
16 
52 

= 
4 
13 

In Experiment 4, the events are nonmutually exclusive. The addition causes the king of
clubs to be counted twice, so its probability must be subtracted. When two events are nonmutually exclusive, a
different addition rule must be used.  
Addition Rule 2: 
When two events, A and B, are nonmutually exclusive, the
probability that A or B will occur is:


P(A or B) = P(A) + P(B)  P(A and B) 
In the rule above, P(A and B) refers to the overlap of the two events. Let's
apply this rule to some other experiments.  
Experiment 5: 
In a math class of 30 students, 17 are boys and 13 are girls. On a unit test,
4 boys and 5 girls made an A grade. If a student is chosen at random from the class, what is the
probability of choosing a girl or an A student?


Probabilities: 
P(girl or A) 
= 
P(girl) 
+ 
P(A) 
 
P(girl and A) 


= 
13 
+ 
9 
 
5 
30 
30 
30 

= 
17 
30 

Experiment 6: 
On New Year's Eve, the probability of a person having a car accident is 0.09. The probability of a person driving while intoxicated is 0.32 and probability of a person having a car accident while intoxicated is 0.15. What is the probability of a person driving while intoxicated or having a car accident?


Probabilities: 
P(intoxicated or accident) 
= 
P(intoxicated) 
+ 
P(accident) 
 
P(intoxicated and accident) 


= 
0.32 
+ 
0.09 
 
0.15 

= 
0.26 


Summary: 
To find the probability of event A or B, we must first determine whether the events are
mutually exclusive or nonmutually exclusive. Then we can apply the appropriate Addition
Rule:


Addition Rule 1: 
When two events, A and B, are mutually exclusive, the
probability that A or B will occur is the sum of the probability of each event.


P(A or B) = P(A) + P(B) 

Addition Rule 2:: 
When two events, A and B, are nonmutually exclusive, there is some
overlap between these events. The probability that A or B
will occur is the sum of the probability of each event, minus the probability of the
overlap. 

P(A or B) = P(A) + P(B)  P(A and B) 

Exercises
Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer
is provided in the RESULTS BOX. If you make a mistake, choose a different button. 
1. 
A day of the week is chosen at random. What is the probability of choosing a Monday or Tuesday?

 


2. 
In a pet store, there are 6 puppies, 9 kittens, 4 gerbils and 7 parakeets. If a pet is chosen at random,
what is the probability of choosing a puppy or a parakeet?

 


3. 
The probability of a New York teenager owning a skateboard is 0.37, of owning a bicycle is 0.81 and of owning both is 0.36. If a New York teenager is chosen at random, what is the
probability that the teenager owns a skateboard or a bicycle? 
 


4. 
A number from 1 to 10 is chosen at random. What is the probability of choosing a
5 or an even number? 
 


5. 
A single 6sided die is rolled. What is the probability of rolling a number greater than 3 or an even
number?

 


Unit 6: Probability Theory
