# 2.1.3.2.4 - Complements

2.1.3.2.4 - Complements

The 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:

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

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