The complementary of an event is that the event does not happen.
For example the event to you role a six faced die and get a 3 has a complementary event that you will get either a 1, 2,4,5, or 6.
complementary event and the event can not happen in the same time.
P( event)+P(complementary event) =1
P( complementary event)= 1-P(event)
If we have to calculate the probability of getting either a 1, 2, 4, 5, or 6 on your next role of a die we can make our life easy and say we have to calculate the probability of getting not getting a 3.
P( 3) = 1/6
P (not getting 3) =1-1/6=5/6