A magician at a magic show says he will throw a deck of cards in the air and pierce a red card, either hearts or diamonds, or a 9. Is this an example of a mutually inclusive event? What is the probability of one of these events occurring?

In this concept, you will learn how to calculate the probability of overlapping, or mutually inclusive events.

### Mutually Inclusive Events

There are some events that can impact each other or that are overlapping. They have one or more outcomes in common. When events have outcomes in common, they are said to be **overlapping events**, also known as **mutually inclusive events**.

Let's look at an example.

For a single spin, are events \begin{align*}R (\text{red})\end{align*} and \begin{align*}T (\text{top})\end{align*} overlapping events?

First, 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

Next, compare the list.

Then, decide if the two events in question have a common outcome.

The two events DO have an outcome in common–red-top.

The answer is the two events do have a common outcome, therefore \begin{align*}R\end{align*} and \begin{align*}T\end{align*} are mutually inclusive events.

So \begin{align*}R\end{align*} and \begin{align*}T\end{align*} are mutually inclusive events**. **Notice that the events have more than one thing in common. They have red in common, but they also have the words “top” in common too.

Let’s look at another example.

For a single toss of a number cube, are events smaller than 6 and greater than 4 disjoint events or overlapping events?

First, make a list of the outcomes.

Smaller outcomes 1, 2, 3, 4, 5,

Greater outcomes: 5, 6

Next, compare the list.

Then, decide if the two events in question have a common outcome.

The two events have 1 outcome in common = 5.

The answer is events smaller than 6, \begin{align*}S6\end{align*}, and greater than 4, \begin{align*}G4\end{align*}, are mutually inclusive events.

To find the probability of mutually inclusive events occurring, the following formula is used:

X = one set of outcomes Y = another set of outcomes, where X and Y are known to have mutually inclusive events:

\begin{align*}P (\text{A or} \ B) = P(A) + P(B) - P (A\ and\ B)\end{align*}

where P(A or B) is the probability tht either event A or event B will occur, P(A) is the probability that event A will occur, P(B) is the probability that event B will occur and P(A and B) is the probability that both events will occur.

For example:

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

even outcomes: 2, 4, 6

> 4 outcomes: 5, 6

When you consider the outcomes, the mutually inclusive outcome, the one that both events have in common, is 6.

\begin{align*}P (\text{even outcomes or} \ greater\ than\ 4) = P(\frac{3}{6}) + P(\frac{2}{6}) - P (\frac{1}{6})\end{align*}

The answer is the probability that a number cube will land on an even number or a number greater than four is \begin{align*}\frac{4}{6} = \frac{2}{3} = 0.67 = 67\ percent\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about the magician and his promise.

He claims he will pierce a red or a 9 out of a deck of cards thrown up into the air. Is this a mutually inclusive event and, if so, what is the probability one of the favorable events will occur?

First, make list of all outcomes:

red cards in a deck outcomes: 1 hearts 1 diamonds, 2, hearts, 2 diamonds, 3 hearts, 3 diamonds, 4, hearts, 4 diamonds, 5 hearts, 5 diamonds, 6 hearts, 6 diamonds, 7 hearts, 7 diamonds, 8 hearts, 8 diamonds, 9 hearts, 9 diamonds, 10 hearts, 10 diamonds, Jack hearts, Jack diamonds, Queen hearts, Queen diamonds, King hearts, King diamonds, Ace heart, Ace diamonds

9 cards in a deck: 9 clubs, 9 spades, 9 diamonds, 9 hearts

Next, consider the list and identify if any mutuallly inclusive events are present:

There are 2 mutuallly inclusive events: 9 diamonds, 9 hearts.

Then, create a ratio of favorable outcomes to total outcomes to indicate the probability of the favorable outcome occurring:

\begin{align*}P (\text{9 or} \ red) = P(\frac{4}{52}) + P(\frac{26}{52}) - P (\frac{2}{52}) = \frac{28}{52} = 0.54 = 54\ percent\end{align*}

The answer is that the events of a red 9 out of a deck of cards being pierced by the magician is mutually inclusive. The probability of this outcome is 1/26.

#### Example 2

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

First, list the odd outcomes, outcomes greater than 3, and total outcomes.

odd outcomes: 1, 3, 5

> 3 outcomes: 4, 5, 6

total outcomes: 1, 2, 3, 4, 5, 6

Next, consider the list and identify the mutually inclusive outcomes:

There is only one mutually inclusive outcome, the favorable outcome, which is 5.

Then, using the formula for mutually inclusive probability, calculate the probability percent:

\begin{align*}P (\text{odd outcomes or} \ greater\ than\ 3) = P(\frac{3}{6}) + P(\frac{3}{6}) - P (\frac{1}{6}) = \frac{5}{6} = 0.83 = 83\ percent\end{align*}

#### Example 3

For a single spin of the spinner above, are \begin{align*}G (\text{green})\end{align*} and \begin{align*}T (\text{top})\end{align*} disjoint events or overlapping events?

First, make a list of the outcomes.

\begin{align*}G (\text{green})\end{align*} outcomes = green-bottom

\begin{align*}T (\text{top})\end{align*} outcomes = red-top, blue-top

Next, compare the list.

Then, decide if the two events in question have a common outcome.

The two events do not have a common outcome.

The answer is \begin{align*}G (\text{green})\end{align*} and \begin{align*}T (\text{top})\end{align*} are disjoint events because they do not have any outcomes in common.

#### Example 4

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

First, list all possible outcomes:

2 outcomes: 2

even outcomes: 2, 4, 6

total outcomes: 2, 4, 6

Next, consider the list and identify the mutually inclusive outcomes:

There is only one mutually inclusive outcome, the favorable outcome, which is 2.

Then, create the ratio of the number of favorable outcomes to total outcomes.

\begin{align*}P (2\ or \ even) = P(\frac{1}{6}) + P(\frac{3}{6}) - P (\frac{1}{6}) = \frac{3}{6} = 0.50 = 50\ percent\end{align*}

The answer is the probability of a number cube landing on a two and an even number is \begin{align*}\frac{1}{6}\end{align*}.

### Review

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.

- What is the probability of pulling out a red scarf?
- What is the probability of pulling out a primary color?
- What is the probability of pulling out a primary color and is either red or blue?
- What is the probability of pulling out a primary color and is either blue or green?
- What is the probability of pulling out a complementary color?
- What is the probability of pulling out a complementary color and is either green or purple?

A number cube numbered 1 - 12 is rolled.

- What is the probability of rolling an even number?
- What is the probability of rolling an odd number?
- What is the probability of rolling an even number or a number less than four?
- What is the probability of rolling an even number or a number greater than 10?
- What is the probability of rolling an odd number or a number greater than 5?
- What is the probability or rolling an odd number or a one?
- What is the probability of rolling an odd number or a number less than 8?
- What is the probability of rolling an even number or a number less than 11?
- What is the probability of rolling an odd number or a number greater than 9?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.5.