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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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Practice Complement Rule for Probability
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Identify Overlapping, Disjoint, and Complementary Events

Have you ever thought about how one event is connected to another event? Take a look at this dilemma.

Mary has a single number cube and a spinner. If she rolls the number cube and then spins the spinner are the events complementary or disjoint?

To answer this question, you will have to know about different types of events. Use this Concept to learn all about disjoint, complementary and overlapping events.

### Guidance

When we spin a spinner or roll a die to calculate a probability, some probabilities have events in common and some don’t. This is where we 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.

• Event A: {yellow}
• Event B: {blue}

Events A and B are disjoint events because they have no outcomes in common – the arrow either lands on blue or yellow.

We can use a Venn diagram to show when events overlap and when they don’t overlap. A Venn diagram is something that you may have seen before. It has round shapes that overlap or don’t overlap. The Venn diagram for disjoint events shows no overlap between the two events.

Not all events are disjoint. There are many events that are connected to each other. Let’s look at this one.

Consider this spinner and the events R (red) and T (top).

• Event R: {red-top, red-bottom}
• Event T: {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.

Complementary events are events whose probability sum adds up to 1 (decimal) or 100 percent.

Events Y and G in this spinner above are complementary.

P(yellow)+P(green)=1\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.

Use the yellow and green spinner to answer the following questions.

#### Example A

If you calculate the yellow part of the spinner as a decimal, what decimal would it be?

Solution: \begin{align*}.25\end{align*}

#### Example B

What would that be as a percent?

Solution: \begin{align*}25\%\end{align*}

#### Example C

What would the green portion be as a decimal?

Solution: \begin{align*}.75\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

Mary's events are disjoint because one outcome will not affect the other outcome.

### Vocabulary

Disjoint events
events that don’t have any outcomes in common.
Complementary events
probability that has a sum of 100%. Either/Or events are complementary events.

### Guided Practice

Here is one for you to try on your own.

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

Solution

These two events are complementary events because 50% of the spinner is red and 50% is blue. Together these two parts add up to one whole or 100%.

### Practice

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

1. For a flip of a coin, are events \begin{align*}H(\text{heads})\end{align*} and \begin{align*}T(\text{tails})\end{align*} disjoint or overlapping?
2. For a flip of a coin, are events \begin{align*}H(\text{heads})\end{align*} and \begin{align*}T(\text{tails})\end{align*} complementary or non-complementary?
3. Why?
4. For a single toss of a number cube, are \begin{align*}E(\text{even})\end{align*} and \begin{align*}T(3)\end{align*} disjoint events or overlapping events?
5. Why?
6. For a single toss of a number cube, are \begin{align*}E(\text{even})\end{align*} and \begin{align*}S(6)\end{align*} disjoint events or overlapping events?
7. For a single toss of a number cube, are \begin{align*}G3(\text{greater than} \ 3)\end{align*} and \begin{align*}O(\text{odd})\end{align*} disjoint events or overlapping events?
8. For a single toss of a number cube, are \begin{align*}E(\text{even})\end{align*} and \begin{align*}O(\text{odd})\end{align*} complementary events or non-complementary events?

1. For a single spin, are \begin{align*}B(\text{blue})\end{align*} and \begin{align*}G(\text{green})\end{align*} disjoint events or overlapping events?
2. For a single spin, are \begin{align*}G(\text{green})\end{align*} and \begin{align*}L(\text{left})\end{align*} disjoint events or overlapping events?
3. For a single spin, are \begin{align*}Y(\text{yellow})\end{align*} and \begin{align*}R(\text{red})\end{align*} complementary or non-complementary events?
4. For a single spin, are \begin{align*}R(\text{right})\end{align*} and \begin{align*}L(\text{left})\end{align*} complementary or non-complementary events?
5. For a light switch, are ON and OFF disjoint or overlapping events?
6. For a light switch, are ON and OFF complementary or non-complementary events?
7. For an oven, are ON and OFF disjoint or overlapping events?

### Vocabulary Language: English

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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