# 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 Section

When flipping a coin, one can flip heads or tails. Thus, $$P(Tails^{C})=P(Heads)$$ and $$P(Heads^{C})=P(Tails)$$

## Example: Hearts Section

If you randomly select a card from a standard 52-card 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: $$P(A^{C})=1-P(A)$$

It is also true then that: $$P(A)=1-P(A^{C})$$

## Example: Rain Section

According to the weather report, there is a 30% chance of rain today: $$P(Rain) = .30$$

Raining and not raining are complements.

$$P(Not \:rain)=P(Rain^{C})=1-P(Rain)=1-.30=.70$$

There is a 70% chance that it will not rain today.

## Example: Winning Section

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 Section

In a standard 52-card 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 Section

Of individuals with two hands, it is possible to be right-handed, left-handed, or ambidextrous. Assuming that these are the only three possibilities and that there is no overlap between any of these possibilities:

$$P(right\;handed)+P(left\;handed)+P(ambidextrous) = 1$$