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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
Credit: Simon Law
Source: https://www.flickr.com/photos/sfllaw/1339777790/in/photolist-33oHg5-dTeZk5-8wcNGV-rdB5RJ-8dVhg2-8Svt9i-9wiPpn-5GKDq8-38Vktr-a9uHGN-bS7PQT-bS7J8p-6X8FuA-6mEgyi-53CBCe-7QywsN-dTeZTw-dT9qdc-ho7tiP-4M5qq9-aeVYbL-qUB1mR-5fyBQw-5fyBAm-4nM1Pj-arQLbg-pig7NP-dvn5px-fQDEzn-33fT32-82x7Qd-nPb9Qw-7pdUVp-8zMLSA-4eapEJ-eitqYZ-nyJhVQ-96mjDW-9jrqCx-56xzis-37XPAX-fro4iC-996gq7-eKYyoV-bS7J2Z-bS7PMK-9dnkzB-kJqUUv-8rcfgG-6hYa7U
License: CC BY-NC 3.0

The middle school’s spring carnival was starting right after school, so Cheryl sat impatiently in class waiting for the bell to ring. Her goal was to be one of the first students in line for the cake walk, which was sort of like playing musical chairs. In the cake walk, 20 students walk around a numbered circle and stop on the closest number when the music stops. The caller then draws a number, and whoever is standing on that number wins a cake of their choice. Cheryl has her heart set on winning a chocolate cake. If Cheryl has a 5% chance of ending up on the selected number, what is the probability that she will not end up on the selected number?

In this concept, you will learn how to find the probability of complementary events.

Finding Probability of Complementary Events

When you know the likelihood that something will happen, you can determine or base your actions on that event happening. If you knew today was going to be a sunny day or a rainy day, and the weather report stated there was a 10% chance of rain, you can figure out the complementary event or the opposite probability, which would be 90% chance of sun. If there is a 10% chance that it will not be sunny, then there is a 90% chance that it will be sunny.

Here is another way to look at complements. In the following statement, \begin{align*}P(A)\end{align*} means the “probability of \begin{align*}A\end{align*}” and \begin{align*}P(B)\end{align*} is the “probability of \begin{align*}B\end{align*}.”

If \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are complements, then \begin{align*}P(B) = 100\% - P(A)\end{align*}. For example, if the probability of \begin{align*}A\end{align*}, \begin{align*}P(A)\end{align*} is 10%, then the probability of \begin{align*}B\end{align*}, \begin{align*}P(B)\end{align*} is \begin{align*}100\% - 10\%\end{align*}, which is 90%.

Here is an example.

Write the complementary event for the probability shown here.

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

First, 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.

Next, write the complementary event, which is the opposite probability.

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

You 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.

Sometimes, when it comes to determining the probability of an event, you just need to use common sense and ask yourself if the event is likely to occur, unlikely to occur, certain to occur, or is impossible. One way to do this is by looking at the complementary event. The more likely the complementary event is, then the less likely the original situation will be.


Example 1

Earlier, you were given a problem about Cheryl and her quest for a chocolate cake.

Cheryl is third in line when Mrs. Mixon and Jackie open the cake walk. She pays her $1 then takes her place on the circle with the other 19 students. There are 20 numbers on the circle, so each student will end up on a number. That means she has a 1 in 20 chance of winning, which is a 5% probability. Cheryl begins to think she just wasted $1when she considers the odds of her losing.

First, consider the opposite probability of Cheryl winning.

She will lose because she won’t end up on the selected number.

Next, subtract the probability she will win, which is 5%, from 100% to get the probability that she will lose.


There is a 95% chance that Cheryl will lose.

Example 2

Write a complementary event for the following situation.

The boy has a 40% chance of winning the race.

First, consider the opposite probability, which is the boy losing the race.

Next, subtract 40% from 100% to get the probability of the boy losing.


There is a 60% chance the boy will lose the race.

Example 3

Write a complementary event for the following situation.

The course has a 74% passing rate.

First, consider the opposite probability, which is the failure rate.

Next, subtract 74% from 100% to get the failure rate.


The course has a 26% failure rate.

Example 4

Write a complementary event for the following situation.

There is a 90% chance that Teddy will pass the driver’s exam on his first try.

First, consider the opposite probability, which is Teddy failing on his first try.

Next, subtract 90% from 100% to get the failure rate.


There is a 10% chance that Teddy will fail on his first try.

Example 5

Write a complementary event for the following situation.

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

First, consider the opposite probability, which is, it won’t snow.

Next, subtract 20% from 100% to get the probability that it won’t snow.


There is an 80% chance that it won’t snow.


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.

Find the complement’s percentage.

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

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

8. \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*}.

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

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

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

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

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

Determine whether the two events are 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

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.16. 


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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.
Disjoint Events Disjoint or mutually exclusive events cannot both occur in a single trial of a given experiment.
enumerate Enumerate means to catalogue or list members independently.
Venn diagrams A diagram of overlapping circles that shows the relationship among members of different sets.

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