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Statistics 101: Principles of Statistics11 chapters | 141 lessons | 9 flashcard sets

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Lesson Transcript

Instructor:
*Cathryn Jackson*

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

In this lesson, you will learn the differences between mutually exclusive and non-mutually exclusive events and how to find the probabilities of each using the Addition Rule of Probability.

Cheyenne and her friends are playing a board game. Each person roles a die on his or her turn and moves the number of spaces indicated on the die. Some of the spaces ask the player to pull a card from a deck of regular playing cards. Cheyenne needs to move three or six spaces to get to the next card space. Then, she needs to pick a black card or a seven. What is the probability of Cheyenne rolling a 3 or a 6 on the die? What is the probability of Cheyenne picking a card that is a black card or a seven?

In this lesson, you will learn about the **Addition Rule of Probability**, which is a rule for finding the union of two events, either mutually exclusive or non-mutually exclusive.

Each of these scenarios represents an event in probability. The first event, rolling the die on either a 3 or a 6, is a **mutually exclusive** event, which are events that cannot happen at the same time.

This Venn diagram represents the mutually exclusive events of rolling a 3 or a 6. Notice that there is no overlap between the two circles because the events cannot happen at the same time.

The second event, picking a card out of a deck that is black or a seven is an example of **non-mutually exclusive** events, which are events that can happen separately or at the same time.

This Venn diagram represents picking either a seven or a black card from the deck.

Notice that there is a place where the two circles overlap. This is because it is possible to pick a black card that is a seven. This is an example of non-mutually exclusive events because the two events can occur at the same time.

Now let's look at how to use the Addition Rule of Probability to find the probability of mutually exclusive and non-mutually exclusive events.

Remember, rolling the die would be an example of a mutually exclusive event. The die cannot land on two sides at the same time; therefore the probability of each side of the die is mutually exclusive. You may also hear of mutually exclusive events called disjoint events. When working with mutually exclusive events in probability, use the following formula:

This formula is read as:

*The probability of event A or B is equal to the probability of event A plus the probability of event B.*

To find the probability of mutually exclusive events, follow these steps:

- Find the total of possible outcomes
- Find the desired outcomes
- Create a ratio for each event
- Add the ratios, or fractions, of each event

First, the total possible outcomes of a six-sided die are six. You have six different possible outcomes when rolling the die.

Second, find the desired outcomes. Cheyenne needs to roll a 3 or a 6. Therefore, a 3 or a 6 would be the desired outcome. A 3 appears on a six-sided die once and so does a 6. Remember this information for the next step.

Third, create a ratio for each event. The first event, rolling a 3 would be a ratio of 1/6, because the die only has one side with three dots. The second event, rolling a 6, would also be a ratio of 1/6, because the die only has one side with six dots.

Fourth, add the ratios, or fractions, of each event. This step will give you the probability of rolling a die and getting a 3 or a 6.

*1/6 + 1/6 = 2/6 or 1/3*

Therefore, Cheyenne has a 1 in 3 chance of rolling a 3 or a 6. Once she rolls a 3 or a 6, Cheyenne can land on a space that allows her to pick a card. She needs to pick a black card, or a seven.

Remember, picking a black card or a seven card out of a deck of regular playing cards is an example of non-mutual events. If you are looking for the probability of two events happening at the same time, this is called the intersection of two events. Learn more about intersection in our Multiplication Rule of Probability lesson. This is the formula for non-mutually exclusive events:

This formula is read as:

*The probability of event A or B is equal to the probability of event A plus the probability of event B minus the probability of event A and B.*

To find the probability of non-mutually exclusive events, follow these steps:

- Find the total of possible outcomes
- Find the desired outcomes
- Create a ratio for each event
- Add the ratios, or fractions, of each event
- Subtract the overlap of the two events

First, the total number of possible outcomes of a deck of regular playing cards is 52, since there are 52 cards in a regular deck.

Second, find the desired outcomes. Cheyenne needs to select a black card or a seven card. Therefore, a black or a 7 card would be the desired outcome. There are two suits that are black cards: spades and clubs. There are 13 cards for each suit. Therefore, the desired outcome of possibilities for a black card is 26. There are four sevens in a regular deck of playing cards, one seven for each suit. Therefore, the desired outcome possibilities for a seven card are 4.

Third, create a ratio for each event. The first event, selecting a black card, would be a ratio of 26/52. The second event, selecting a seven card, would be a ratio of 4/52. I got these ratios by using the desired outcome number as the numerator and the total possible outcomes as the denominator.

Fourth, add the ratios, or fractions, of each event like this:

*26/52 + 4/52 = 30/52*

You might be inclined to stop here and say that there is a 30 in 52 chance of picking a black card or a seven card. But, notice that we have the word 'or' in that statement. That means that you are not looking for a card that is a black seven, just all of the cards that are black and a seven. Therefore, you need to subtract the overlap of the two events in the probability. The probability of getting a black seven is 2/52 because there are only two black sevens in the deck. Take this ratio and subtract from the previous probability like this:

*30/52 - 2/52 = 28/52*

Now we have the correct probability. Cheyenne has a 28 in 52 chance of selecting a card that is either a black or a seven. I'd say those are pretty good odds!

Example 1:

*Abby is attending her first swimming competition. There are seven girls racing in the first heat. She has to place first or second to make it to the next level of the tournament. Assuming there are no ties, what is the probability Abby will get first or second?*

Abby has a 2 in 7 or approximately 29% chance of making it into the next level of the tournament.

This is another example of mutually exclusive events. Abby can't get both first and second place. Therefore, there is no overlap of events. Because this is an example of mutually exclusive events, we can use this formula from the Addition Rule of Probability:

Abby has a 1/7 chance of getting first place, and a 1/7 chance of getting second place. We can add these two probabilities together to find the probability of Abby getting first or second like this:

*1/7 + 1/7 =2/7*

Example 2:

*Abby's team ranks first out of the other teams at the end of the swimming competition. The team goes out for pizza and ice cream afterward. There are 20 people on the team; 8 people order pizza, and 12 people order ice cream. Out of the team, 5 people ended up ordering both pizza and ice cream. What is the probability of a team member ordering pizza or ice cream, but not both?*

The probability of a team member ordering pizza or ice cream, but not both, is 15 out of 20 or 75%.

This is an example of non-mutually exclusive events since some of the team members were able to order both ice cream and pizza. The probability of a team member ordering pizza is 8/20 since we were already given that information. The probability of a team member ordering ice cream is 12/20. First we can add those two probabilities together:

*8/20 + 12/20 = 20/20*

You've probably decided at this point there is something wrong, since there are only twenty people on the team. That's because at some point there is an overlap in the numbers. Remember, some people ordered both pizza and ice cream. We know from the problem that 5 people ordered both pizza and ice cream. We need to subtract that probability 5/20 from our problem like this:

*20/20 - 5/20 = 15/20*

Remember that probability is estimation or a prediction in this case. We are trying to predict whether or not a team member would actually order both, or one or the other. Therefore, we can only accurately say, that there were 5 people that ordered both. We can say if a teammate does not order both, there is a 75% chance he or she will order one or the other.

The **Addition Rule of Probability** is a rule for finding the union of two events: either mutually exclusive or non-mutually exclusive. **Mutually exclusive** events are events that cannot happen at the same time. **Non-mutually exclusive** events are events that can happen separately or at the same time.

To find the union of two events that are mutually exclusive, use this formula:

The probability of event A or B is equal to the probability of event A plus the probability of event B.

To find the union of two events that are non-mutually exclusive, use this formula:

The probability of event A or B is equal to the probability of event A plus the probability of event B minus the probability of event A and B.

Remember, the Addition Rule of Probability helps you find the probability of event A or event B, not both events. To find the intersection of two events, check out our lesson on the Multiplication Rule of Probability.

Once this lesson is completed, you should be able to:

- Recall the Addition Rule of Probability
- Compare/contrast a mutually exclusive event to a non-mutually exclusive event and give an example
- Remember the formulas for calculating the probability of a non-mutually exclusive or a mutually exclusive event
- Calculate the probability of a mutually exclusive or non-mutually exclusive event

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Statistics 101: Principles of Statistics11 chapters | 141 lessons | 9 flashcard sets

- Mathematical Sets: Elements, Intersections & Unions 3:02
- Events as Subsets of a Sample Space: Definition & Example 4:51
- Probability of Simple, Compound and Complementary Events 6:55
- Probability of Independent and Dependent Events 12:06
- Probability of Independent Events: The 'At Least One' Rule 5:27
- How to Calculate Simple Conditional Probabilities 5:10
- The Relationship Between Conditional Probabilities & Independence 7:52
- Using Two-Way Tables to Evaluate Independence 8:09
- Applying Conditional Probability & Independence to Real Life Situations 12:32
- The Addition Rule of Probability: Definition & Examples 10:57
- Math Combinations: Formula and Example Problems 7:14
- How to Calculate a Permutation 6:58
- How to Calculate the Probability of Permutations 10:06
- Relative Frequency & Classical Approaches to Probability 5:56
- Go to Probability

- Go to Sampling

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