<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

Mutually Inclusive Events

Probability of two events that can occur at the same time P(A or B) = P(A) + P(B) -  P(A and B)

Atoms Practice
Estimated15 minsto complete
%
Progress
Practice Mutually Inclusive Events
Practice
Progress
Estimated15 minsto complete
%
Practice Now
Mutually Inclusive Events

Let's Think About It

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

Guidance

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 R(red) and T(top) overlapping events?

First, make a list of the outcomes.

R outcomes: red-top, red bottom

T 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 R and T are mutually inclusive events.

So R and T 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, S6, and greater than 4,  G4, 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:

P(A or B)=P(A)+P(B)P(A and B) 

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.

 P(even outcomes or greater than 4)=P(36)+P(26)P(16)  

The answer is the probability that a number cube will land on an even number or a number greater than four is 46=23=0.67=67 percent.

Guided Practice

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:

P(odd outcomes or greater than 3)=P(36)+P(36)P(16)=56=0.83=83 percent 

Examples

Example 1

For a single spin of the spinner above, are G(green) and T(top) disjoint events or overlapping events?

First, make a list of the outcomes.

G(green) outcomes = green-bottom

T(top) 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 G(green) and T(top)  are disjoint events because they do not have any outcomes in common.

Example 2

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.

 P(2 or even)=P(16)+P(36)P(16)=36=0.50=50 percent

The answer is the probability of a number cube landing on a two and an even number is  16.

Follow Up

 

Remember 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:

 P(9 or red)=P(452)+P(2652)P(252)=2852=0.54=54 percent

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.

Video Review

Explore More

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?

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 12.5. 

Vocabulary

Addition Principle

Addition Principle

If events A and B are mutually inclusive, then P(A or B) = P(A) + P(B) – P(A and B)
Mutually Exclusive Events

Mutually Exclusive Events

Mutually exclusive events have no common outcomes.
Mutually Inclusive Events

Mutually Inclusive Events

Mutually inclusive events can occur at the same time.

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Mutually Inclusive Events.

Reviews

Please wait...
Please wait...

Original text