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The Addition Rule of Probability: Definition & Examples

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  • 0:01 Addition Rule
  • 1:48 Mutually Exclusive Events
  • 3:51 Non-Mutually Exclusive Events
  • 7:00 Practice Problems
  • 9:46 Lesson Summary
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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.

Addition Rule

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.

Mutually exclusive events have no overlap
venn diagram of mutually exclusive event

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.

Non-mutually exclusive events can have an overlap
venn diagram of non-mutually exclusive event

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.

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:

Formula for calculating the probability of a mutually exclusive event
formula for a mutually exclusive event

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:

  1. Find the total of possible outcomes
  2. Find the desired outcomes
  3. Create a ratio for each event
  4. 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.

Non-Mutually Exclusive Events

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:

Formula for calculating the probability of non-mutually exclusive events
Formula for calculating the probability of 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:

  1. Find the total of possible outcomes
  2. Find the desired outcomes
  3. Create a ratio for each event
  4. Add the ratios, or fractions, of each event
  5. 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

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