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# Complement Rule for Probability

## If P(A) is is the probability that event A happens then the probability that event A doesn't happen is 1-P(A).

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Complement Rule for Probability

Have you ever played a basketball game at an amusement park? Well, Jeff is going to do just that.

Jeff stopped going on rides to play a basketball game at the amusement park. In this game, Jeff had ten chances to make a basket. Kyle stopped by to support Jeff in his attempt.

"I have to make four out of ten," Jeff told Kyle.

"What do you think your chances are?"Kyle asked.

"Well, I'm pretty good at basketball. I bet I have a 75% chance of getting the four in."

If Jeff has a 75% chance of making the basket, what is the probability that he won't make the basket?

This is an situation with complementary events. You will learn how to answer this question during Concept.

### Guidance

What happens when we know the likelihood that something will happen? Well, we can determine or base our actions on that event happening.

If there is a 10% chance of rain, what is the probability that it will be sunny? We can say that there is a 90% chance that it will be sunny.

If someone only knew that there was a 10% chance that it would be rainy, and that if it wasn't rainy the only other option was to be sunny, could they tell the chance of it being sunny? To figure this out, we have to figure out what the chances are of something not happening. This is called a complementary event.

If there is a 10% chance that it will not be sunny, then there is a 90% chance that it will be sunny.

Write the complementary event for the probability shown here.

There is a 50% chance that Mary will be coming over on Saturday.

To write the complementary event, we look at the opposite probability. There is a 50% chance that Mary will be coming over, so there is a 50% chance that she will not be coming over.

There is a 50% chance that Mary will not be coming over.

We can write complementary events as fractions, decimals and percents. Use whatever form is used in the example and have the complementary event match that form.

We can also predict how likely an event is to happen or not to happen based on common sense. Some things we can know for certain and some things are left up to chance.

The sunrise is one of those events. We know that the sun will rise tomorrow. Sometimes we won’t see it due to weather, but it certainly will rise.

You can also catch yourself arguing about this too. How do we know that it will rise? You could find yourself debating this with a friend for a long time. However, we need to use common sense when we are thinking about these things and not just figuring the probability using numbers.

Don’t get too caught up!

Predict whether each event is likely, impossible, unlikely or certain or write a complementary event for each situation.

#### Example A

The team lost its last four games, it is __________ that they will win tonight.

Solution: Unlikely

#### Example B

On her fifth birthday, Joanna turned five years old.

Solution: Joanna is five years old now.

#### Example C

There is a 20% chance that it will snow tonight.

Solution: There is an 80% chance that it will not snow.

Here is the original problem once again.

Jeff stopped going on rides to play a basketball game at the amusement park. In this game, Jeff had ten chances to make a basket. Kyle stopped by to support Jeff in his attempt.

"I have to make four out of ten," Jeff told Kyle.

"What do you think your chances are?"Kyle asked.

"Well, I'm pretty good at basketball. I bet I have a 75% chance of getting the four in."

If Jeff has a 75% chance of making the basket, what is the probability that he won't make the basket?

To answer this question, we use 100% as our total and subtract 75% from 100%. This is the part of the whole not represented in the first percent.

10075=25\begin{align*}100 - 75 = 25\end{align*}

Jeff has a 25% chance of not getting the basket.

### Vocabulary

Here are the vocabulary words in this Concept.

Probability
the chances that something will happen. It can be written as a fraction, decimal or percent.
Ratio
compares two quantities. In probability the ratio compares the number of favorable outcomes to the number of possible outcomes
Complementary Events
For every probability that something will happen, there is a probability that it won’t happen. These two ratios are complementary events.

### Guided Practice

Here is one for you to try on your own.

Use likely, not likely or uncertain to describe the following event.

Mary will eat chocolate ice cream this week.

This is an uncertain event. We haven't been given any other information about Mary and her likes and dislikes when it comes to ice cream. Therefore, this is uncertain.

### Video Review

Here is a video for review.

### Practice

Directions: Use common sense and make a prediction, use likely, impossible unlikely or certain to describe each statement.

1. Our team has a perfect record. It is _________ that we will win on Saturday.

2. A baby born will either be a boy or a girl.

3. A pig will fly through the sky.

4. A cat will like a dog.

5. There is an 85% chance it will rain. It is ________ that it will rain.

Directions: Find the complement.

6. A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are complements. Missing \end{align*}\begin{align*}P (B) = 15%\end{align*}. Find P(A)\begin{align*}P (A)\end{align*}.

7. C\begin{align*}C\end{align*} and D\begin{align*}D\end{align*} are complements. Missing \end{align*}\begin{align*}P (C) = 80%\end{align*}. Find P(D)\begin{align*}P (D)\end{align*}.

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

9. T\begin{align*}T\end{align*} and S\begin{align*}S\end{align*} are complements. Missing \end{align*}\begin{align*}P (T) = 33%\end{align*}. Find P(S)\begin{align*}P (S)\end{align*}.

10. L\begin{align*}L\end{align*} and K\begin{align*}K\end{align*} are complements. Missing \end{align*}\begin{align*}P (K) = 70%\end{align*}. Find P(L)\begin{align*}P (L)\end{align*}.

11. A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*} are complements. Missing \end{align*}\begin{align*}P (B) = 12%\end{align*}. Find P(A)\begin{align*}P (A)\end{align*}.

12. N\begin{align*}N\end{align*} and M\begin{align*}M\end{align*} are complements. Missing \end{align*}\begin{align*}P (N) = 96%\end{align*}. Find P(M)\begin{align*}P (M)\end{align*}.

13. Q\begin{align*}Q\end{align*} and Z\begin{align*}Z\end{align*} are complements. Missing \end{align*}\begin{align*}P (Q) = 10%\end{align*}. Find P(Z)\begin{align*}P (Z)\end{align*}.

Directions: Write complementary or not complementary.

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

15. Percentage of votes that 2 candidates get in a 3-candidate election

16. Winning a game or losing a game

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### Vocabulary Language: English

complement

A mutually exclusive pair of events are complements to each other. For example: If the desired outcome is heads on a flipped coin, the complement is tails.

Complement rule

The Complement Rule states that the sum of the probabilities of an event and its complement must equal 1, or for the event A, P(A) + P(A') = 1.

enumerate

Enumerate means to catalogue or list members independently.

Probability

Probability is the chance that something will happen. It can be written as a fraction, decimal or percent.

Venn diagrams

A diagram of overlapping circles that shows the relationship among members of different sets.