

If A is an event within the sample space S of an experiment, the complement of A (which may be written as A^{C}) consists of all outcomes in S that are not in A.


The complement of A is everything else in the sample space that is NOT in A.
Consider these experiments where an event and its complement are stated:
Experiment: Tossing a Coin 
Event 
A 
The coin shows heads. 
Complement 
A^{C} 
The coin shows tails. 
Experiment: Drawing a Card
The cards in a standard deck are either red or black. 
Event 
A 
The card is black 
Complement 
A^{C} 
The card is red 

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

Since the sum of probabilities of all possible events equals 1, the probability that event A will not occur is equal to 1 minus the probability that event A will occur.
A pair of dice are rolled. What is the probability of not rolling doubles? 
There are 6 ways to roll doubles: (1,1), (2,2), (3,3), (4,4), (5,5) (6,6).
P(doubles) = 6/36 = 1/6
P(not doubles) = 1  1/6 = 5/6 
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 
A gum ball machine contains gum balls of five different colors: 36 red, 44 white, 15 blue, 20 green, and 5 orange. The machine dispenser randomly selects one gum ball. What is the probability that the gum ball selected is:
a.) green?
b.) not green?
c.) not orange?
d.) orange?
e.) red, white, or blue?
f.) not a color in the flag of the USA? 

There are 120 gum balls 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.) the probability of red, white or blue is 36/120 + 44/120 + 15/120 = 95/120 = 19/24.
f.) use the complement from part e: 1  19/24 = 5/24. 
NOTE: The reposting of materials (in part or whole) from this site to the Internet
is copyright violation
and is not considered "fair use" for educators. Please read the "Terms of Use". 
