Lesson 5
Math 3201
Mutually Exclusive Events
We will use Venn diagrams for illustrating unions and intersection of sets
applying it to the Principle of Inclusion and Exclusion for determining the
probability of non-mutually exclusive events.
We are familiar with classifying events that are mutually exclusive(disjoint
sets) and events that are non-mutually exclusive.
We worked with Venn diagrams and represented mutually exclusive sets as
sets that did not intersect(i.e., n A  B  0 ).
We are also familiar with an event and its complement. Remember the
outcomes of an event and the outcomes of the complement make up the
entire sample space.
We will now solve probability problems that involve non-mutually exclusive
events and mutually exclusive events. It is important to not only apply the
formula for the probability of two non-mutually events, but to understand
how it is developed.
For example; D represents students on the debate team and B represents
students on the basketball team.
D
B
40
12
27
19
We should recognize that in order to determine the probability of events,
they have to think about the following questions:

Are the two sets intersecting or disjoint?
Intersecting

How many elements are in each set?
n(D) = 52

n(B) = 39
How many elements are in the universal set S?
n(U) = 98
Using the Principle of Inclusion and Exclusion and the probability for nonmutually exclusive events:
n  D  B   n D   n  B   n D  B 
n D   n  B   n  D  B 
P D  B  
nS 
n D  n B  n  D  B 
P D  B  


nS  nS 
nS 
P D  B   P D   P B   P D  B 
If the events are mutually exclusive, then nD  B  0 , therefore for
disjoint sets, PD  B  PD  PB .
Example:
Pearl is about to draw a card at random from a standard deck
of 52 playing cards. If she draws a face card or a spade, she
will win a point.
A)
B)
C)
A)
Draw a Venn diagram to represent the two events.
Are the events mutually exclusive?
Determine the probability of drawing a face card or a
spade.
Let F represent face cards.
Let S represent spades.
Let U represent universal set.
U
F
Jh,Qh,Kh,
Jd,Qd,Kd,
Jc,Qc,Kc
Js
Qs
Ks
As,2s,3s
4s,5s,6s
7s,8s,9s,
10s
S
Ah,2h,3h,4h,5h,6h,7h,8h,9h,10h,
Ad,2d,3d,4d,5d,6d,7d,8d,9d,10d,
Ac,2c,3c,4c,5c,6c,7c,8c,9c,10c
B)
C)
The two events are not mutually exclusive because they share the
Jack of spades, Queen of spades and the King of spades.
PF  S   PF   PS   PF  S 
12 13 3

 
52 52 52
12  13  3

52
22 11


or
42%
52 26
Worksheet 4
1.
2.
Math 3201
Name:___________
Classify the events in each experiment as either mutually exclusive or
non-mutually exclusive:
A)
The experiment is rolling a die. The first event is rolling an even
number and the second event is rolling a prime number.
B)
The experiment is selecting a gift. The first event is that the
gift is edible and the second event is that the gift is an iPhone.
Zach is playing a board game. He must roll two four-sided dice,
numbered 1 to 4. He can move if he rolls a sum of 2 or a sum of 8.
A)
Use A and B to represent the two events that will allow Zach to
move. Then draw a Venn diagram to illustrate A and B.
B)
Are A and B mutually exclusive or not mutually exclusive.
C)
Determine the probability that Zach will roll a sum of 2 or a
sum of 8.
D)
Determine the probability that Zach will roll doubles or a sum
of 6.
3.
For each of the following, state whether the events are mutually
exclusive. Explain your reasoning.
A)
Selecting a prime number or selecting an even number from a
set of 15 balls, numbered 1 to 15.
B)
Rolling a sum of 10 or a sum of 7 with a pair of six-sided dice,
numbered 1 to 6.
C)
Walking to school or getting a ride to school.
4.
Tanya plays the balloon pop game at a carnival. There are 40 balloons,
with the name of a prize inside each balloon. The prizes are 8 stuffed
bears, 5 toy trucks, 16 decks of cards, 7 yo-yos, and 4 giant stuffed
dogs. Tanya pops a balloon with a dart. Determine the odds in favour
of her winning either a stuffed dog or a stuffed bear.
6.
The probability that John will study on Friday night is 0.4.
The probability that he will play video games on Friday night is 0.6.
The probability that he will do at least one of these activities is 0.8.
A)
Determine the probability that he will do both activities.
B)
Are these events mutually exclusive? Explain how you know.