2.1.3.2.4  Complements
2.1.3.2.4  Complements Complement

The probability that the event does not occur. The complement of \(P(A)\) is \(P(A^C)\). This may also be written as \(P(A')\).
In the diagram below we can see that \(A^{C}\) is everything in the sample space that is not A.
 Complement of A
 \(P(A^{C})=1−P(A)\)
Example: Coin Flip
When flipping a coin, one can flip heads or tails. Thus, \(P(Tails^{C})=P(Heads)\) and \(P(Heads^{C})=P(Tails)\)
Example: Hearts
If you randomly select a card from a standard 52card deck, you could pull a heart, diamond, spade, or club. The complement of pulling a heart is the probability of pulling a diamond, spade, or club. In other words: \(P(Heart^{C})=P(Diamond,\; Spade,\;\;Club)\)
The complement of any outcome is equal to one minus the outcome. In other words: \(P(A^{C})=1P(A)\)
It is also true then that: \(P(A)=1P(A^{C})\)
Example: Rain
According to the weather report, there is a 30% chance of rain today: \(P(Rain) = .30\)
Raining and not raining are complements.
\(P(Not \:rain)=P(Rain^{C})=1P(Rain)=1.30=.70\)
There is a 70% chance that it will not rain today.
Example: Winning
The probability that your team will win their next game is calculated to be .45, in other words:
\(P(Winning)=.45\)
Winning and losing are complements of one another. Therefore the probability that they will lose is:
\(P(Losing)=P(Winning^{C})=1.45=.55\)
The sum of all of the probabilities for possible events is equal to 1.
Example: Cards
In a standard 52card deck there are 26 black cards and 26 red cards. All cards are either black or red.
\(P(red)+P(black)=\frac{26}{52}+\frac{26}{52}=1\)
Example: Dominant Hand
Of individuals with two hands, it is possible to be righthanded, lefthanded, or ambidextrous. Assuming that these are the only three possibilities and that there is no overlap between any of these possibilities:
\(P(right\;handed)+P(left\;handed)+P(ambidextrous) = 1\)