Section 3.3
Addition Rule
(Venn Diagram)
Section 3.3 Objectives
• Determine if two events are mutually exclusive
• Use a Venn Diagram to find the probability of two
events (uses the Addition Rule)
Mutually Exclusive Events
Mutually exclusive
• Two events A and B cannot occur at the same time
A
B
A and B are mutually
exclusive
A
B
A and B are not mutually
exclusive
Example: Mutually Exclusive Events
Decide if the events are mutually exclusive.
Event A: Roll a 3 on a die.
Event B: Roll a 4 on a die.
Solution:
Mutually exclusive (The first event has one outcome, a
3. The second event also has one outcome, a 4. These
outcomes cannot occur at the same time.)
Example: Mutually Exclusive Events
Decide if the events are mutually exclusive.
Event A: Randomly select a male student.
Event B: Randomly select a nursing major.
Solution:
Not mutually exclusive (The student can be a male
nursing major.)
The Addition Rule
Addition rule for the probability of A or B
The probability that events A or B will occur for
mutually exclusive events A and B is
P(A or B) = P(A) + P(B)
(Can be extended to any number of mutually
exclusive events)
Example: Using the Addition Rule
(mutually exclusive events)
The frequency distribution shows
the volume of sales (in dollars)
and the number of months in
which a sales representative
reached each sales level during
the past three years. If this sales
pattern continues, what is the
probability that the sales
representative will sell between
$75,000 and $124,999 next
month? ( 36 months)
Sales volume ($)
Months
0–24,999
3
25,000–49,999
5
50,000–74,999
6
75,000–99,999
7
100,000–124,999
9
125,000–149,999
2
150,000–174,999
3
175,000–199,999
1
Solution: Using the Addition Rule
• A = monthly sales between
$75,000 and $99,999
• B = monthly sales between
$100,000 and $124,999
• A and B are mutually exclusive
P( A or B)  P( A)  P( B)
7
9


36 36
16

 0.444
36
Sales volume ($)
Months
0–24,999
3
25,000–49,999
5
50,000–74,999
6
75,000–99,999
7
100,000–124,999
9
125,000–149,999
2
150,000–174,999
3
175,000–199,999
1
Non- Mutually Exclusive Events
D – event of being on Dean’s list
A – event of being an athlete
P(D) = 0.25
P(A) = 0.20
P(A and D) = 0.05
What is Probability of being an Athlete and not being on
the Dean’s list? P(A and not D)
A and D are ~not~ mutually exclusive
(someone can be an athlete and be on
Dean’s list!)
A
D
P(A and D) = .05 (5%)
A
D
5
P(A) = 20
A
15
D
5
P(D) = 25
A
15
D
5
20
Total Probability is 100%
A
15
D
5
20
60
P(A and not D) = 15
A
15
D
5
20
60
P(not A) = 20+60 = 80
P(not D) = 15+60 = 75
P(A’) = 1 – P(A) = 100 – 20 = 80
P(D’) = 1 – P(D) = 100 – 25 = 75
A
15
D
5
20
60
P(A and D) = 5
P(A or D) = 15 + 5 + 20 = 40
A
15
D
5
20
60
P(A or D, but not both) = 15 + 20 = 35
P(not A and not D) = 60
A
15
D
5
20
60
P(A and not D) = 15
P(D and not A) = 20
A
15
D
5
20
60
Non- Mutually Exclusive Events
M – event of being male
E – event of being younger than 18
P(M) = 79%
P(E) = 18%
P(M and E) = 14%
P(M and E) = 14
M
E
14
P(M) = 79
P(E) = 18
M
65
E
14
4
Total probability is 100%
M
65
E
14
4
17
P(not M) = 4+17 = 21
[ same as P(M’) ]
P(not E ) = 65+17 = 82
[ same as P(E’) ]
P(M or E) = 83 P(M and not E) = 65 P(E and not M) = 4
P(not M and not E) = 17
P(M or E, but not both) = 69
M
65
E
14
4
17
P(not M) = 4+17 = 31
[ same as P(M’) ]
P(not E ) = 65+17 = 82
[ same as P(E’) ]
P(M or E) = 83 P(M and not E) = 65 P(E and not M) = 4
P(not M and not E) = 17
P(M or E, but not both) = 69
M
65
E
14
4
17
Section 3.3 Summary
• Determined if two events are mutually exclusive
• Used the Addition Rule (using Venn diagram) to find
the probability of two events