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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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Identify Overlapping, Disjoint, and Complementary Events

Let’s Think About It

Credit: Artform Canada
Source: https://www.flickr.com/photos/artform/3191659950/in/photolist-aqRPD2-dmAbcS-5S36iQ-4QrZxN-856iH9-wSiwW-6yrGH-ctWHA5-7MQkE-7KsGxA
License: CC BY-NC 3.0

Instead of staying home to watch cartoons on Saturday morning, Stella decided to go with her dad to his squash club. However, when she got there she found there wasn’t much to do but look at all the different squash players. So Stella decided to make a game of it. Stella has black hair and blue eyes, so she decided to count how many of the people she saw also had those characteristics. By the end of her dad’s game, Stella had counted 27 members in total. Of those 27, 19 had black hair, 14 had blue eyes, and 11 had both black hair and blue eyes. How can Stella tell if these events are disjoint or overlapping? 

In this concept, you will learn to identify and distinguish between overlapping, disjoint and complementary events.

Guidance

When you spin a spinner or roll a die to calculate a probability, some probabilities have events in common and some don’t. This is where you can begin to talk about identifying disjoint events. Disjoint events are events that don’t have any outcomes in common.

Take a look at this situation.

Consider spinning this spinner:

License: CC BY-NC 3.0

  • Event \begin{align*}A\end{align*}: {yellow} 
  • Event \begin{align*}B\end{align*}: {blue}

Events \begin{align*}A\end{align*} and \begin{align*}B\end{align*} are disjoint events because they have no outcomes in common - the arrow either lands on blue or yellow.

You can use a Venn diagram to show when events overlap and when they don’t overlap. A Venn diagram shows a way of representing whether events have any outcomes in common. It has round shapes that overlap or don’t overlap. The Venn diagram for disjoint events shows no overlap between the two events.

License: CC BY-NC 3.0

Not all events are disjoint. There are many events that are connected to each other.

Let’s look at another example.

Consider this spinner and the events \begin{align*}R\end{align*} (red) and \begin{align*}T\end{align*} (top).

License: CC BY-NC 3.0

  • Event \begin{align*}R\end{align*}: {red-top, red-bottom }
  • Event \begin{align*}T\end{align*}: {red-top, blue-top }

Clearly, both events share an outcome  red-top  so the two are called overlapping events. The Venn diagram for overlapping events shows that the two events overlap, or share 1 or more outcomes.

License: CC BY-NC 3.0

Complementary events are events whose probability sum adds up to 1.0 or 100 percent.

License: CC BY-NC 3.0

Events \begin{align*}Y\end{align*} and \begin{align*}G\end{align*} in this spinner above are complementary.

\begin{align*}P(\text{yellow})+P(\text{green})=1\end{align*}

Complementary events are either-or events. Either the spinner above lands on green or it lands on yellow. There are no other outcomes.

Guided Practice

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Are event red and event blue complementary or disjoint events? Why?

First, remember the definitions of overlapping and complementary events.

Overlapping events are events that have outcomes in common.

Complementary events are events whose probability sum adds up to 1 or 100%.

Next, look at the spinner. 50% of the spinner is red and 50% is blue. There is no section on the spinner that has both red and blue.

The answer is complementary. Therefore, the two events are complementary.

Examples

Example 1

For the spinner below, if event \begin{align*}A\end{align*} is bottom and event \begin{align*}B\end{align*} is blue, are the events disjoint or overlapping. Are the events complementary?

License: CC BY-NC 3.0

First, remember the definitions of disjoint and overlapping events.

Disjoint events are events that don’t have any outcomes in common.

Overlapping events are events that have outcomes in common.

Complementary events are events whose probability sum adds up to 1 or 100%.

Next, look at the spinner to determine what type of event is occurring.

Event \begin{align*}A\end{align*} is the sections with the word bottom.

Event \begin{align*}B\end{align*} includes the blue sections.

Then determine whether these events are complementary, disjoint or overlapping.

They are not complementary because the red - top is not included in the two events.

They are not disjoint as there is a blue - bottom.

Therefore the events are overlapping.

Example 2

If event \begin{align*}A\end{align*} includes the numbers 1, 6, 7, and 9 and event \begin{align*}B\end{align*} includes the numbers 2, 3, 4, 5, and 8. Is this an example of disjoint or overlapping events?

First, let’s list all of the numbers in Events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}.

Event \begin{align*}A\end{align*}: 1, 6, 7, 9.

Event \begin{align*}B\end{align*}: 2, 3, 4, 5, 8.

Next, remember the definitions of disjoint and overlapping events.

Disjoint events are events that don’t have any outcomes in common.

Overlapping events are events that have outcomes in common.

Then, determine whether the events are disjoint or overlapping. You can do this by drawing a Venn diagram that shows if any events are common to the two sets.

License: CC BY-NC 3.0

The answer is that these events are disjoint as there are no numbers common to both events.

Example 3

If Event \begin{align*}A\end{align*} is the set of even numbers from 0 to 20 and Event \begin{align*}B\end{align*} is the set of numbers that are multiples of 4 and less than 30, Is this an example of disjoint or overlapping events?

First, let’s list all of the numbers in Events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}.

Event \begin{align*}A\end{align*}: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20.

Event \begin{align*}B\end{align*}: 4, 8, 12, 16, 20, 24, 28.

Next, remember the definitions of disjoint and overlapping events.

Disjoint events are events that don’t have any outcomes in common.

Overlapping events are events that have outcomes in common.

Then, determine whether the events are disjoint or overlapping. You can do this by drawing a Venn diagram that shows if any events are common to the two sets.

License: CC BY-NC 3.0

The answer is that these events are overlapping as there are numbers common to both events.

Follow Up

Credit: Michael Pedersen
Source: https://www.flickr.com/photos/46289172@N04/4429959240/in/photolist-aqRPD2-dmAbcS-5S36iQ-4QrZxN-856iH9-wSiwW-6yrGH-ctWHA5-7MQkE-7KsGxA
License: CC BY-NC 3.0

Remember bored Stella at the squash club?

Stella needs to determine if the events of black hair and blue eyes are disjoint or overlapping.

First, let’s list all of the numbers in Events \begin{align*}A\end{align*} and \begin{align*}B\end{align*}.

Event \begin{align*}A\end{align*}: black hair

Event \begin{align*}B\end{align*}: 4, blue eyes

Next, remember the definitions of disjoint and overlapping events.

Disjoint events are events that don’t have any outcomes in common.

Overlapping events are events that have outcomes in common.

Then, determine whether the events are disjoint or overlapping. You can do this by drawing a Venn diagram that shows if any events are common to the two sets.

Remember that there are 19 people with black hair and 11 also have blue eyes, therefore 8 have eyes of a different color.

Also, 14 people have blue eyes and 11 of these also have black hair, therefore 3 have hair of a different color.

License: CC BY-NC 3.0

Looking at the Venn diagram above, it is clear that these events are overlapping as they have outcomes in common.

Video Review

https://www.youtube.com/watch?v=obZzOq_wSCg

Explore More

Solve the problems. For overlapping events, tell which events overlap.

1. For a flip of a coin, are events \begin{align*}H\end{align*} (heads) and \begin{align*}T\end{align*} (tails) disjoint or overlapping?

2. For a flip of a coin, are events \begin{align*}H\end{align*} (heads) and \begin{align*}T\end{align*} (tails) complementary or non-complementary?

3. Why?

4. For a single toss of a number cube, are \begin{align*}E\end{align*} (even) and \begin{align*}T\end{align*} (3) disjoint events or overlapping events?

5. Why?

6. For a single toss of a number cube, are \begin{align*}E\end{align*} (even) and \begin{align*}S\end{align*} (6) disjoint events or overlapping events?

7. For a single toss of a number cube, are \begin{align*}G3\end{align*} (greater than 3) and \begin{align*}O\end{align*} (odd) disjoint events or overlapping events?

8. For a single toss of a number cube, are \begin{align*}E\end{align*} (even) and \begin{align*}O\end{align*} (odd) complementary events or non-complementary events?

License: CC BY-NC 3.0

9. For a single spin, are \begin{align*}B\end{align*} (blue) and \begin{align*}G\end{align*} (green) disjoint events or overlapping events?

10. For a single spin, are \begin{align*}G\end{align*} (green) and \begin{align*}L\end{align*} (left) disjoint events or overlapping events?

11. For a single spin, are \begin{align*}Y\end{align*} (yellow) and \begin{align*}R\end{align*} (red) complementary or non-complementary events?

12. For a single spin, are \begin{align*}R\end{align*} (right) and \begin{align*}L\end{align*} (left) complementary or non-complementary events?

13. For a light switch, are ON and OFF disjoint or overlapping events?

14. For a light switch, are ON and OFF complementary or non-complementary events?

15. For an oven, are ON and OFF disjoint or overlapping events?

Vocabulary

complement

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.
Complementary Events

Complementary Events

Complementary events can occur in a single trial of a given experiment.
Complement rule

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 Events

Disjoint or mutually exclusive events cannot both occur in a single trial of a given experiment.
enumerate

enumerate

Enumerate means to catalogue or list members independently.
Venn diagrams

Venn diagrams

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

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