#### Objective

Here you will learn how to identify the complement of a given theoretical or experimental probability.

#### Concept

What does it mean to find the ** complement** of an event? Why would you want to do so?

This lesson is all about what this lesson is not about, so stick about and we’ll figure it out!

#### Watch This

http://youtu.be/SOHm-dSJRl0 TenMarksInstructor – Complement of an Event - Probability

#### Guidance

The ** complement** of an event is the sample space of all outcomes that are

*not*the event in question. The complement of the event “a flipped coin lands on heads” is “a flipped coin lands on tails”. The complement of “A six-sided die lands on 1 or 2”. Is “A six-sided die lands on 3, 4, 5, or 6”. Complements are notated using the prime symbol ’ as in: \begin{align*}P(A^\prime)\end{align*} is the complement of \begin{align*}P(A)\end{align*}.

The probability of the complement of an event is always whatever probability it would take to reach 100%. If the probability of pulling a green marble out of a bag is 26%, then the probability of the compliment (pulling a *not* green marble) is 74%.

By convention, we most often see probabilities described as either a percentage or a fraction. Despite this, every calculated or experimental probability can be expressed as a value between 0 and 1 since percentages and fractions can all be converted to decimals and a probability must be between 0% and 100%. For example, a probability of \begin{align*}\frac{3}{4}\end{align*} or 75% could also be expressed as the decimal .75, and a probability of 20% or \begin{align*}\frac{1}{5}\end{align*} could be expressed as the decimal .20.

One benefit of viewing probabilities as decimals is that it is easy to calculate the ** complementary** probability of a given event by subtracting the event probability (expressed as a decimal) from 1.

**Example A**

What is the complement to the event “Brian chooses one of the 2 red shirts from his drawer containing 10 shirts”?

**Solution:** The complement would be the other possibility: “Brian chooses one of the *not* red shirts from his drawer.”

**Example B**

If the probability of randomly choosing a Queen from a standard deck of 52 cards is .077, what is the probability of the complementary event?

**Solution:** The complement would be choosing a card that is *not* a Queen, and the complement probability would be the difference between .077 and 1:

\begin{align*}P(Queen^\prime)=1-.077=.923\end{align*}

Therefore, the if the probability of choosing a Queen is 7.7%, then the probability of choosing a card *not* a Queen is 92.3%

**Example C**

What is the probability of the compliment of the event: “Roll a standard die and get an even number”?

**Solution:** There are three even numbers on a standard die: 2, 4, and 6. That means that the probability that you *do* roll and get an even number is:

\begin{align*}P(even)=\frac{3}{6} \ or \ 50 \%\end{align*}

Therefore, the complement is:

\begin{align*}P(even^\prime)=1-.50=.5 \ or \ 50 \%\end{align*}

**Concept Problem Revisited**

The complement of an event is the set of all outcomes that are *not* the event.

#### Vocabulary

The ** complement** of an event is notated using the prime symbol ’ such as: “The complement of \begin{align*}P(A)\end{align*} is \begin{align*}P(A^\prime)\end{align*}”. \begin{align*}P(A^\prime)\end{align*} is the sample space of all outcomes not a part of \begin{align*}P(A)\end{align*}, and can be calculated as \begin{align*}1-P(A)\end{align*}.

#### Guided Practice

- If \begin{align*}P(X)=\frac{1}{6}\end{align*}, what is \begin{align*}P(X^\prime)\end{align*}?
- What is the probability of the complement to a probability of 74%?
- What is the complement to the event: “flipped coin lands on heads”?
- What is the probability of the complement of randomly choosing one of the 3 quarters from a set of 10 coins?
- What is the percent probability of \begin{align*}Y^\prime\end{align*} if \begin{align*}\text{P(Y)}=\frac{1}{8}\end{align*}?

**Solutions:**

- \begin{align*}P(X^\prime)=1-P(X)=1-\frac{1}{6}=\frac{5}{6}\end{align*}
- The complement probability is \begin{align*}100 \% -74 \%=26 \%\end{align*}
- “flipped coin lands on tails”
- The event probability is \begin{align*}\frac{3 \ quarters}{10 \ coins}=\frac{3}{10}\end{align*}. The complement is \begin{align*}1-\frac{3}{10}=\frac{7}{10}\end{align*}
- \begin{align*}P(Y^\prime)=1-P(Y)=1-12.5 \%=87.5 \%\end{align*}

#### Practice

For problems 1 – 10, identify the **percent probability** of the **complement** of the described event.

- Roll a standard die once and get an even number.
- Pull a red card from a standard deck.
- Pull a face card from a standard deck.
- Roll two standard dice and get a sum greater than 9.
- Pull two cards from a deck, without replacement, get at least one face card.
- Roll a 10-sided die twice, get a 6 both times.
- The probability that a student in your class likes chocolate is 34%.
- Of the 76 students in your math class, 26 earned an A.
- 23% of million-mile cars are Toyotas.
- A candy machine has 24 green, 32 red, and 14 yellow candies in it. You choose a yellow candy.
- There are 150 students in your class, 40 have laptops, and 110 have tablets. 26 of those students have both a laptop and a tablet. What is the probability that a randomly chosen student has a tablet, given that she has a laptop?
- Roll two standard dice, and get 4’s on both, given that you know that you have already rolled a 4 on one of them.
- Draw two cards in a row, without replacement, that are the same suit from a standard deck.
- Roll of two standard dice once, getting a sum greater than 8, given that one of the dice is a 6.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 6.7.