Have you ever tried to calculate probability using a number cube? Take a look at this situation.

Andy tosses a number cube.

What is the probability that the number that lands up will be both even and less than 6?

**In this Concept, you will learn how to calculate the probability of overlapping events. Pay attention and we will revisit this dilemma at the end of the Concept.**

### Guidance

Previously we worked on disjoint events, now 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.**

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 scenario.

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, 6

Greater outcomes: 4, 5, 6

**Step 2:** The two events have 1 outcome in common = 6.

**\begin{align*}S\end{align*} and \begin{align*}G4\end{align*} are overlapping events.**

Answer the following questions about overlapping events.

#### Example A

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?

**Solution: Green and top are disjoint events because they do not have any outcomes in common.**

#### Example B

A number cube is tossed. What is the probability that it will be a two and an even number?

**Solution: \begin{align*}\frac{1}{6}\end{align*}**

#### Example C

A number cube is tossed. What is the probability that it will be an even number and a number greater than four?

**Solution: \begin{align*}\frac{1}{6}\end{align*}**

Here is the original problem once again.

Andy tosses a number cube.

What is the probability that the number that lands up will be both even and less than 6?

Let's start by listing out the possible total outcomes.

\begin{align*}1, 2, 3, 4, 5, 6\end{align*}

Now let's list out the even numbers.

\begin{align*}2, 4, 6\end{align*}

Next we can list the numbers less than 6.

\begin{align*}1, 2, 3, 4, 5\end{align*}

There are two numbers that overlap.

\begin{align*}\frac{2}{6} = \frac{1}{3}\end{align*}

**This is our probability.**

### Vocabulary

- 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.

### Guided Practice

Here is one for you to try on your own.

A number cube is tossed. What is the probability that the number that lands up will be both odd and greater than 3?

**Answer**

**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.

Notice that there is only one overlapping outcome. This is the favorable outcome.

\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.**

### Video Review

This is a James Sousa video on probability.

### Practice

Directions: Write the probability of each overlapping event occurring.

Eight colored scarves are put into a bag. They are red, yellow, blue, green, purple, orange, brown and black.

1. What is the probability of pulling out a red scarf?

2. What is the probability of pulling out a primary color?

3. What is the probability of pulling out a primary color and is either red or blue?

4. What is the probability of pulling out a primary color and is either blue or green?

5. What is the probability of pulling out a complementary color?

6. What is the probability of pulling out a complementary color and is either green or purple?

A number cube numbered 1 - 12 is rolled.

7. What is the probability of rolling an even number?

8. What is the probability of rolling an odd number?

9. What is the probability of rolling an even number or a number less than four?

10. What is the probability of rolling an even number or a number greater than 10?

11. What is the probability of rolling an odd number or a number greater than 5?

12. What is the probability or rolling an odd number or a one?

13. What is the probability of rolling an odd number or a number less than 8?

14. What is the probability of rolling an even number or a number less than 11?

15. What is the probability of rolling an odd number or a number greater than 9?