The simplest example is flipping a coin: heads and tails are mutually exclusive in that, if the head side appears, the tail side won't, and vice versa.
A die is being rolled. Find the probability p(3 or 5) 


Since a die cannot show both a 3 or a 5 at the same time, these events are mutually exlusive.
p(3 or 5)

=

p(3)+p(5)


Probability of mutually events


=



Substitution 

=



Add 
The probability of rolling a 3 or a 5 is or about 33%
The Cost Less Clothing Store carries remainder pairs of slacks. Is you buy a pair of slacks in your regular waist size without trying them on, the probability that the waist will be too tight is 0.30 and the probability that it will be too loose is 0.10.
(a) Are the events "too tight" and "too loose" mutually exclusive?
The waist cannot be both too tight and too loose at the same time, so the events are mutually exclusive.
(b) If you choose a pair of slacks at random in your regular waist size, what is the probability that the waist will be too tight or too loose?
Since the events are mutually exclusive,
p(too tight or too loose) 
= 
p(too tight)+p(too loose) 

= 
0.30+0.10 

= 
0.40 