# 12.6: Complement Rule for Probability

Difficulty Level: At Grade Created by: CK-12
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Practice Complement Rule for Probability

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Have you ever watched a horse race?

Lucky Seven is a beautiful horse in the Kentucky Derby. Many people say he is a favorite to win. Lucky Seven might win the race or he might not. He has a 69% chance of winning.

What is the probability that he won't win the race?

Winning or losing a race is an complementary event. Pay attention in this Concept and you will be able to figure out this unknown probability.

### Guidance

In an earlier Concept, disjoint events are either-or events. When you flip a coin you either flip heads or you flip tails.

Similarly, for this spinner the events \begin{align*}R (\text{red})\end{align*} and \begin{align*}B (\text{blue})\end{align*} are disjoint events. The probability of one of the two disjoint events occurring is just the sum of the probabilities of the events.

\begin{align*}P (\text{red or blue}) &= P (\text{red}) + P (\text{blue})\\ &= \frac{1}{2} + \frac{1}{2} = 1\end{align*}

In other words, one of the two events must occur. The probability of either red or blue is 1. The arrow will land on either red or blue 100 percent of the time.

When one of two events must occur the two events are said to be complementary. The sum of the probabilities of two complementary events adds up to 1 or 100 percent of the outcomes of the events.

Here are some situations which involve complementary events.

• Flipping a coin heads or flipping a coin tails.
• Turning on a light switch on or turning a light switch off.
• Locking a door or unlocking a door.

Though some complementary events are “50-50” events, such as flipping a coin, not all complementary events are “50-50.”

For the spinner shown:

\begin{align*}P (\text{blue or yellow}) &= P (\text{blue}) + P (\text{yellow})\\ &= \frac{3}{4} + \frac{1}{4}\\ &= 1\end{align*}

For the spinner above, the events \begin{align*}B (\text{blue})\end{align*} and \begin{align*}Y (\text{yellow})\end{align*} are complementary because their probabilities add up to 1. But the two complements are not equal in size.

Note that some disjoint events are NOT complementary events. Here, \begin{align*}R (\text{red})\end{align*} and \begin{align*}B (\text{blue})\end{align*} are disjoint events. However, their probabilities do NOT add up to 1 or 100 percent:

\begin{align*}P (\text{red or blue}) &= P (\text{red}) + P (\text{blue})\\ &= \frac{1}{4} + \frac{1}{4}\\ &= \frac{1}{2}\end{align*}

Since the sum of any two complements is 1, if you know the probability of one complement, you can find the probability of the other.

For events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, suppose the probability of \begin{align*}B\end{align*} is 0.4. That means:

\begin{align*}P (A) + P (B) &= 1\\ P (A) + 0.4 &= 1\end{align*}

You should be able to guess that the probability of \begin{align*}P(A)\end{align*} is 0.6, because:

\begin{align*}P (A) + P (B) &= 1\\ 0.6 + 0.4 &= 1\end{align*}

An easier way to find a complement is to use the following rules.

Complement Rule: For any two complements, \begin{align*}A\end{align*} and \begin{align*}B\end{align*}, the value of \begin{align*}P (A) = 1 - P (B)\end{align*}. In practical terms:

Subtract the complement you know from 1 to find an unknown complement.

\begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complements. \begin{align*}P (B) = 0.3\end{align*}. Find \begin{align*}P (A)\end{align*}.

To figure this out, subtract the complement you know, 0.3, from 1 to find \begin{align*}P (B)\end{align*}

\begin{align*}P (B) &= 1 - P (A)\\ &= 1 - 0.3\\ &= 0.7\end{align*}

You can also learn to find probabilities and make predictions.

Disjoint events are events that have no outcomes in common. The events \begin{align*}R (\text{red})\end{align*} and \begin{align*}B (\text{blue})\end{align*} for this spinner are disjoint events.

Overlapping events are events that have one or more outcomes in common. The events \begin{align*}R (\text{red})\end{align*} and \begin{align*}L (\text{left})\end{align*} for the spinner are overlapping events because they have the outcome red-left in common.

Complementary events are a pair of disjoint events whose probability sum adds up to 1. The events \begin{align*}R (\text{red})\end{align*} and \begin{align*}B (\text{blue})\end{align*} for this spinner are complementary events because the sum of their probabilities is 1:

\begin{align*}P (\text{red or blue}) &= P (\text{red}) + P (\text{blue})\\ &= \frac{1}{2} + \frac{1}{2}\\ &= 1\end{align*}

What is the probability that arrow will land on red, green, or yellow?

The events are disjoint so the probability of one of them occurring is the sum of their individual probabilities.

\begin{align*}P (\text{red or blue or green}) &= P (\text{red}) + P (\text{blue}) + P (\text{green})\\ &= \frac{1}{4} + \frac{1}{4} + \frac{1}{4}\\ &= \frac{3}{4}\end{align*}

We can find the probability of an event when we add the probabilities together.

The probability of the Mets winning tonight’s game is 0.6. Predict how likely it is for the Mets to lose tonight’s game.

Winning the game and losing the game are complementary events. So you can use the rule:

\begin{align*}P (\text{lose}) &= 1 - P (\text{win})\\ &= 1 - 0.6\\ &= 0.4\end{align*}

We can write this answer as 40%. There is a 40% chance that the Mets will win tonight.

Here are a few problems for you to try on your own.

#### Example A

A and C are complements. If C is .67, find A.

Solution: \begin{align*}.33\end{align*}

#### Example B

If the Yankees have a 45% chance of winning tonight, what is the probability that they won't win?

Solution: \begin{align*}55%\end{align*}

#### Example C

D and E are complements. If D is .2, what is E?

Solution:\begin{align*}.8\end{align*}

Here is the original problem once again.

Lucky Seven is a beautiful horse in the Kentucky Derby. Many people say he is a favorite to win. Lucky Seven might win the race or he might not. He has a 69% chance of winning.

What is the probability that he won't win the race?

Complementary events have a sum of 1 or 100%. We know the probability of Lucky Seven winning. It is 69%. We can subtract this from 100 to find the probability of him losing.

\begin{align*}100 - 69 = 31\end{align*}

Lucky Seven has a 31% chance of losing the race.

### Vocabulary

Here are the vocabulary words in this Concept.

Disjoint Events
events that are not connected. One outcome does not affect the other.
Complementary Events
One of two events must occur then the two are complementary events. We can subtract one event from 1 to get the other event.

### Guided Practice

Here is one for you to try on your own.

Y and Z are complements. If the probability of Y occurring is 14%, what is the probability of Z occurring?

To figure this out, first you have to remember that complementary events have a sum of 1 or 100%. We know that Y is 14%. We can subtract from 1 or 100% to find Z.

\begin{align*}100 - 14 = 86\end{align*}

The probability of Z occurring is 86%.

### Video Review

Here is a video for review.

### Practice

Directions: Find the complement.

1. \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complements. \begin{align*}P (B) = 0.15\end{align*}. Find \begin{align*}P (A)\end{align*}.

2. \begin{align*}C\end{align*} and \begin{align*}D\end{align*} are complements. \begin{align*}P (C) = 0.8\end{align*}. Find \begin{align*}P (D)\end{align*}.

3. \begin{align*}G\end{align*} and \begin{align*}H\end{align*} are complements. \begin{align*}P (H) = 49\%\end{align*}. Find \begin{align*}P (G)\end{align*}.

4. \begin{align*}T\end{align*} and \begin{align*}S\end{align*} are complements. \begin{align*}P (T) = \frac{3}{8}\end{align*}. Find \begin{align*}P (S)\end{align*}.

5. \begin{align*}L\end{align*} and \begin{align*}K\end{align*} are complements. \begin{align*}P (K) = 0.07\end{align*}. Find \begin{align*}P (L)\end{align*}.

6. \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complements. \begin{align*}P (B) = 0.125\end{align*}. Find \begin{align*}P (A)\end{align*}.

7. \begin{align*}N\end{align*} and \begin{align*}M\end{align*} are complements. \begin{align*}P (N) = 96.1\%\end{align*}. Find \begin{align*}P (M)\end{align*}.

8. \begin{align*}Q\end{align*} and \begin{align*}Z\end{align*} are complements. \begin{align*}P (Q) = \frac{1}{5}\end{align*}. Find \begin{align*}P (Z)\end{align*}.

Directions: Write complementary or not complementary.

9. Percentage of votes that 2 candidates get in a 2-candidate election

10. Percentage of votes that 3 candidates get in a 3-candidate election

11. Winning a game or losing a game

12. Choosing an odd or even number

13. Choosing a number between 1 and 5

14. Passing or failing a test

15. Choosing a color of paint

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