If A is an event within the sample space S of an activity or experiment, the complement of A (denoted A') consists of all outcomes in S that are not in A. 
 The complement of A is everything else in the problem that is NOT in A.

Consider these experiments where an event and its complement are shown:

Experiment:  Tossing a coin
Event

A

The coin shows heads.
Complement

A'

The coin shows tails.
Experiment:  Drawing a card
Event

A

The card is black.
Complement

A'

The card is red.

 

The probability of the complement of an event is one minus the probability of the event.

Complement:

P
(A') = 1 - P(A)

 

Example 1:  A pair of dice are rolled.  What is the probability of not rolling doubles?

There are 6 ways to roll doubles.
P(doubles) = 6/36 = 1/6
P(not doubles) = 1 - 1/6 = 5/6

 

Example 2:  A pair of dice are rolled.  What is the probability of rolling 10 or less?

The complement of rolling "10 or less" is rolling 11 or 12.
P(10 or less) = 1 - P(11 or 12)
= 1 - [P(11) + P(12)]
= 1 - (2/36 + 1/36) = 33/36 = 11/12


 

Example 3:  A gumball machine contains gumballs of five different colors:  36 red, 44 white, 15 blue, 20 green, and 5 orange.  The machine dispenser randomly selects one gumball.  What is the probability that the gumball selected is:
a.) green?
b.) not green?
c.) not orange?
d.) orange?
e.) not a color in the flag of the USA?
f.) red, white, or blue?

There are 120 gumballs in total in the machine.
a.) the probability of green is 20/120 = 1/6.
b.) the probability of not green is 1 - 1/6 = 5/6.
c.) the probability of not orange is 1 - P(orange) = 1 - 5/120 = 1 - 1/24 = 23/24.
d.) the probability of orange is 1/24.
e.) find part f first and then use the complement. 1 - 19/24 = 5/24.
f.) the probability of red, white or blue is 36/120 + 44/120 + 15/120 = 95/120 = 19/24.