# 2.1.3.2.4 - Complements

2.1.3.2.4 - ComplementsThe **complement** of an event is the probability that the event does not occur. The complement of \(P(A)\) is written as \(P(A^C)\) or \(P(A')\).

In the diagram below, we can see that \(A^{C}\) is everything in the sample space that is not A.

Mathematically, if we know \(P(A)\), we can use that value to compute \(P(A^{C})\) using the following formula.

- 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 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: \(P(Heart^{C})=P(Diamond,\; Spade,\;\;Club)\)

## Example: Rain

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

Raining and not raining are complements.

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

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

The sum of all of the probabilities for possible events is equal to 1.

## Example: Cards

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

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:

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