12.3: Complementary Events
Introduction
Juggling and Balancing
Now that Tyler knows the probability of successfully juggling four balls for three minutes, he is certain that this is one of the things that he will do for the talent show. After all, there is an 85% probability that he will be successful. Those are very good odds.
Tyler also knows how to balance a pole on his chin. He can do this for 4 minutes without dropping it. Tyler does a similar experiment as he did with the juggling balls and discovers that he has a 75% probability of being able to balance the pole for four minutes. He is a bit concerned about his odds.
“I am not sure if I should do anything,” he tells Liz after figuring out the probability.
“What do you mean?”
“I mean that if my balancing is that off, maybe I shouldn’t do the juggling either.”
“Why not? Juggling and balancing are disjoint events,” Liz said smiling.
Disjoint events? Tyler isn’t sure what she means. He stops to think about this for minute.
Do you know what Liz means? This lesson is all about different events and their relationships to each other. When we look at events, knowing how they are related can help us in our work. Take some time to learn about disjoint events and then we’ll revisit this problem at the end of the lesson.
What You Will Learn
By the end of this lesson, you will be able to demonstrate the following skills.
- Recognize disjoint events as having no outcomes in common.
- Recognize overlapping events as having one or more outcomes in common.
- Recognize complementary events as two disjoint events, one or the other of which must occur.
- Find probabilities and make predictions involving overlapping, disjoint and complementary events.
Teaching Time
I. Recognize Disjoint Events as Having No Outcomes in Common
Your Aunt Betsy is having a baby! As a mathematician, you realize that this is an opportunity to explore disjoint events. Disjoint events are two events that have no outcomes in common. Here are two events that have no outcomes in common.
- Event B: Having a baby boy.
- Event G: Having a baby girl.
Disjoint events are either-or events.
For example, consider flipping a coin. The two events–flipping heads or flipping tails–have no outcomes in common. You either flip heads or you flip tails.
The probability of one of the two disjoint events occurring is just the sum of the probabilities of the events. Since the probability of flipping heads is \begin{align*}\frac{1}{2}\end{align*} and the probability of flipping tails is \begin{align*}\frac{1}{2}\end{align*}:
\begin{align*}P (\text{heads or tails}) &= P (\text{heads}) + P (\text{tails})\\ &= \frac{1}{2} + \frac{1}{2}\\ &= 1\end{align*}
In other words, all possibilities are covered. The probability of either heads or tails is 1–it will be either heads or tails 100 percent of the time.
Let’s look at an example so that we can better understand disjoint events.
Example
For a single spin, are events \begin{align*}R (\text{red})\end{align*} and \begin{align*}B (\text{blue})\end{align*} disjoint events?
Step 1: To see if \begin{align*}P (\text{red})\end{align*} and \begin{align*}P (\text{blue})\end{align*} or disjoint events, make a list of the outcomes of \begin{align*}P (\text{red})\end{align*} and \begin{align*}P (\text{blue})\end{align*}:
\begin{align*}R\end{align*} outcomes: red
\begin{align*}B\end{align*} outcomes: blue
Step 2: Now compare the list. If there are no outcomes in common, the two events are disjoint.
\begin{align*}R\end{align*} and \begin{align*}B\end{align*} are disjoint events because they have nothing in common.
12E. Lesson Exercises
Consider each pair of events. Tell whether these are disjoint events meaning that they have no connection to each other.
- At a baseball game, striking out and scoring a home run during the same turn at bat.
- Eating an ice cream cone and the sun shining
- Wearing sneakers and being barefoot at the same time.
Check your answers with a friend.
II. Recognize Overlapping Events as Having One or More Outcomes in Common
Now that you have an idea what a disjoint event is, you can begin to think about all sorts of outcomes. You may think that all events are disjoint, but this is not the case. There are some events that impact each other or that are overlapping. They have one or more outcomes in common. Think about this question.
Are all events disjoint events? Not at all, consider the next problem.
Example
For a single spin, are events \begin{align*}R (\text{red})\end{align*} and \begin{align*}T (\text{top})\end{align*} disjoint events?
Step 1: Make a list of the outcomes.
\begin{align*}R\end{align*} outcomes: red-top, red bottom
\begin{align*}T\end{align*} outcomes: red-top, blue-top
Step 2: Compare the list. The two events DO have an outcome in common–red-top. So:
\begin{align*}R\end{align*} and \begin{align*}T\end{align*} are NOT disjoint events.
When events have outcomes in common, they are said to be overlapping events. So:
\begin{align*}R\end{align*} and \begin{align*}T\end{align*} are overlapping events.
How are they overlapping?
Notice that the events have more than one thing in common. They have color in common, but they also have the words “top” or “bottom” in common too. Therefore, the events are overlapping events.
Let’s look at another example.
Example
For a single toss of a number cube, are events Smaller than 6 and Greater than 4 disjoint events or overlapping events?
Step 1: Make a list of the outcomes.
Smaller outcomes 1, 2, 3, 4, 5
Greater outcomes: 5, 6
Step 2: The two events have 1 outcome in common = 5.
\begin{align*}S6\end{align*} and \begin{align*}G4\end{align*} are overlapping events.
12F. Lesson Exercises
Solve this problem.
1. For a single spin, are \begin{align*}G (\text{green})\end{align*} and \begin{align*}T (\text{top})\end{align*} disjoint events or overlapping events?
Discuss your answer with a peer and then continue with the next section.
III. Recognize Complementary Events as Two Disjoint Events, One or the Other of Which Must Occur
As you saw in the first section, 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 examples of 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 example, 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 example, 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.
Example
Problem: \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*}.
Solution: 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*}
IV. Find Probabilites and Make Predictions Involving Overlapping, Disjoint and Complementary Events
Let’s think about the key ideas/concepts of this lesson.
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 the spinner shown above 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 shown above 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 the spinner shown above 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*}
Example
What is the probability that the arrow will land on red, green, or yellow?
Solution: 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.
Example
A number cube is tossed. What is the probability that the number that lands up will be both odd and greater than 3?
Step 1: List the odd outcomes, outcomes greater than 3, and total outcomes. Mark the overlapping outcomes (if they exist).
odd outcomes: 1, 3, 5
> 3 outcomes: 4, 5, 6
total outcomes: 1, 2, 3, 4, 5, 6
Step 2: Find the ratio of favorable outcomes to total outcomes.
\begin{align*}P (\text{odd and} \ >3) = \frac{favorable \ outcomes}{total \ outcomes}=\frac{1}{6}\end{align*}
Notice that we can find the ratio by combining the overlapping outcomes.
Let's try another example:
Example
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.
Solution: 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.
Real–Life Example Completed
Juggling and Balancing
Here is the original problem once again. Reread it and then look at Liz’s explanation of disjoint events.
Now that Tyler knows the probability of his successfully juggling four balls for three minutes, he is certain that this is one of the things that he will do for the talent show. After all, he has an 85% probability that he will be successful. Those are very good odds.
Tyler also knows how to balance a pole on his chin. He can do this for 4 minutes without dropping it. Tyler does a similar experiment as he did with the juggling balls and discovers that he has a 75% probability of being able to balance the pole for four minutes. He is a bit concerned about his odds.
“I am not sure if I should do anything,” he tells Liz after figuring out the probability.
“What do you mean?”
“I mean that if my balancing is that off, maybe I shouldn’t do the juggling either.”
“Why not? Juggling and balancing are independent events,” Liz said smiling.
“What do you mean?” Tyler asks.
“Independent events are events where the outcome of one event does not affect the results of the other event. How you did juggling does not impact how you do balancing. You can still accomplish one or both of them. How you do in each of the events is separate. Does that make sense?”
“Yes. I think I will try both even though my probability of successfully balancing is less than successfully juggling,” Tyler decides.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Disjoint Events
- events that are not connected. One outcome does not affect the other.
- Overlapping Events
- events that have outcomes in common.
- 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.
Time to Practice
Directions: Use what you have learned to solve each problem.
1. For a single toss of a number cube, are events \begin{align*}F (\text{four})\end{align*} and \begin{align*}E (\text{even})\end{align*} disjoint events or overlapping events?
2. For a single toss of a number cube, are \begin{align*}T (\text{three})\end{align*} and \begin{align*}E (\text{even})\end{align*} disjoint events or overlapping events?
3. For a single toss of a number cube, are \begin{align*}O (\text{odd})\end{align*} and \begin{align*}E (\text{even})\end{align*} disjoint events or overlapping events?
4. For a single toss of a number cube, are \begin{align*}T (\text{two})\end{align*} and \begin{align*}L5\end{align*} (less than 5) disjoint events or overlapping events?
5. For a single toss of a number cube, are \begin{align*}S (\text{six})\end{align*} and \begin{align*}O (\text{one})\end{align*} disjoint events or overlapping events?
6. For a single toss of a number cube, are \begin{align*}F (\text{five})\end{align*} and \begin{align*}L5\end{align*} (less than 5) disjoint events or overlapping events?
7. For a single spin, are \begin{align*}R (\text{red})\end{align*} and \begin{align*}Y (\text{yellow})\end{align*} disjoint events or overlapping events?
8. For a single spin, are \begin{align*}R (\text{red})\end{align*} and \begin{align*}L (\text{left})\end{align*} disjoint events or overlapping events?
9. For a single spin, are \begin{align*}P (\text{yellow})\end{align*} and \begin{align*}P (\text{right})\end{align*} disjoint events or overlapping events?
10. For a single spin, are \begin{align*}R (\text{right})\end{align*} and \begin{align*}L (\text{left})\end{align*} disjoint events or overlapping events?
11. For a single spin, are \begin{align*}L (\text{left})\end{align*} and \begin{align*}G (\text{green})\end{align*} disjoint events or overlapping events?
12. For a baby, are \begin{align*}B (\text{boy})\end{align*} and \begin{align*}R (\text{right-handed})\end{align*} disjoint events or overlapping events?
13. For a baby, are \begin{align*}L (\text{left-handed})\end{align*} and \begin{align*}R (\text{right-handed})\end{align*} disjoint events or overlapping events?
14. For a baby, are \begin{align*}G (\text{girl})\end{align*} and \begin{align*}B (\text{brown hair})\end{align*} disjoint events or overlapping events?
15. For a baby, are \begin{align*}G (\text{girl})\end{align*} and \begin{align*}H (\text{heavier than 8 pounds})\end{align*} disjoint events or overlapping events?
Directions: Find the complement.
16. \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*}.
17. \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*}.
18. \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*}.
19. \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*}.
20. \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*}.
21. \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*}.
22. \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*}.
23. \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.
24. Percentage of votes that 2 candidates get in a 2-candidate election
25. Percentage of votes that 2 candidates get in a 3-candidate election
26. Winning a game or losing a game
27. Choosing an odd or even number
28. Choosing a number between 1 and 5
29. Passing or failing a test
30. Choosing a color of paint
Directions: Solve each problem.
31. A six-sided number cube is tossed. What is the probability that the number that lands up will be both even and less than 6?
32. Twenty percent of the customers at Willie’s Pizza Park order pepperoni pizza. Thirty-five percent order cheese pizza. Predict how likely it is that the next customer will order a cheese or pepperoni pizza.
33. The weather report states that there is a 25 percent chance of rain tomorrow. Predict how likely it is that it will not rain tomorrow.
34. A dryer contains 2 thick black socks, 2 thin black socks, and 2 thick white socks. What is the probability that if you pull a sock out of the dryer randomly, it will be black and thick?
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