Addition Rules for Probability

suits

Experiment: A single 6-sided die is rolled. What is the probability of rolling a 2 or a 5?

dice

  

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 6-sided 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?

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

Marbles jarExperiment 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 non-mutually exclusive.

Poker handExperiment 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 non-mutually exclusive. The addition causes the king of clubs to be counted twice, so its probability must be subtracted. When two events are non-mutually exclusive, a different addition rule must be used.

Additional Rule 2: When two events, A and B, are non-mutually 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.

A grade girlExperiment 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

Car CrashExperiment 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 non-mutually 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 non-mutually 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?
   
 

None of the above.

RESULTS BOX:
 

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?
   
 
1

None of the above.

RESULTS BOX:
 

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?
   
 
1.18
0.7
0.82
None of the above.

RESULTS BOX:
 

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

All of the above.

RESULTS BOX:
 

5.  A single 6-sided die is rolled. What is the probability of rolling a number greater than 3 or an even number?
   
 
1

None of the above.

RESULTS BOX: