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

## Chance that either or both events occur using the Addition Rule.

Estimated5 minsto complete
%
Progress
Progress
Estimated5 minsto complete
%
Probability of Unions

On any given night, the probability that Nick has a cookie for dessert is 10%. The probability that Nick has ice cream for dessert is 50%. The probability that Nick has a cookie or ice cream is 55%. What is the probability that Nick has a cookie and ice cream for dessert?

### Probability of Unions

Consider a sample space with events  and .

Recall that the union of events  and  is an event that includes all the outcomes in either event , event , or both. The symbol  represents union. Below,  is shaded.

How do you find the number of outcomes in a union of events? If you find the sum of the number of outcomes in event  and the number of outcomes in event , you will have counted some of the outcomes twice. In fact, you will have counted the outcomes that are in both event  and event  twice. Therefore, in order to correctly count the number of outcomes in the union of two events, you must count the number of outcomes in each event separately and subtract the number of outcomes shared by both events (so these are not counted twice). Generalizing to probability:

. This is called the Addition Rule for Probability.

Note that  is the intersection of the two events. It contains all the outcomes that are shared by both events and is the intersection of the two circles in the Venn diagram.

Suppose that in your class of 30 students, 8 students are in band, 15 students play a sport, and 5 students are both in band and play a sport. Let  be the event that a student is in band and let  be the event that a student plays a sport. Create a Venn diagram that models this situation.

In order to fill in the Venn diagram, remember that the total of the numbers in circle  must be 8 and the total of the numbers in circle B must be 15. The intersection of the two circles must contain a 5.

is the probability that a student is in band or plays a sport or both. With the help of the Venn diagram, this is not too difficult to calculate:

You could also compute this probability using the Addition Rule:

Note that by using the Addition Rule, you avoid having to determine that there are 3 people who are in band and don't play a sport and 10 people who play a sport but are not in band. The Addition Rule is easier when you have not created a Venn diagram.

#### Understanding Probability

Two events  and  are disjoint. Explain why .

If two events are disjoint (also known as mutually exclusive), then they share no outcomes. Therefore, the probability of both events occurring simultaneously is 0 . By the Addition Rule:

#### Real-World Application: Weather Probability

Suppose that today there is a 90% chance of snow, a 20% chance of a strong winds, and a 15% chance of both snow and strong winds. What is the probability of snow or strong winds?

There is a 95% chance of either snow, strong winds, or both.

Now suppose that today there is a 60% chance of snow, an 85% chance of snow or strong winds, and a 25% chance of snow and strong winds. What is the chance of strong winds?

Once again you can use the Addition Rule, because it relates the probabilities in the problem.

There is a 50% chance of strong winds.

### Examples

#### Example 1

Earlier, you were asked what is probability of Nick having a cookie and ice cream for dessert is.

On any given night, the probability that Nick has a cookie for dessert is 10%. The probability that Nick has ice cream for dessert is 50% . The probability that Nick has a cookie or ice cream is 55%. What is the probability that Nick has a cookie and ice cream for dessert?

Let  be the event that Nick has a cookie and let I be the event that Nick has ice cream. The given probabilities are:

You are looking for . By the Addition Rule, you know that . Substitute in the values you know in order to solve for .

The probability that Nick has a cookie and ice cream is 5%.

Consider the experiment of rolling a pair of dice. There are 36 outcomes in the sample space. You are interested in the sum of the numbers. Let  be the event that the sum is even and let B be the event that the sum is less than 5.

#### Example 2

Create a Venn diagram that models this situation. The Venn diagram should contain 36 numbers.

Find all 36 outcomes, and then find the sum of each pair of numbers. Sort the numbers into the Venn diagram so that even numbers are in circle  and numbers less than 5 are in circle . Any other numbers should appear outside of the circles.

#### Example 3

Find  using the Venn diagram.

There are 20 numbers within the circles and 36 numbers total. Therefore, .

#### Example 4

Find  using the Addition Rule and explain why the answer makes sense.

To use the Addition Rule, you need to know  and

Now, use the Addition Rule: . This answer is the same as the answer to #2, as it should be. This calculation makes sense because both  and  include the 4 numbers in the intersection of the circles. You need to subtract , the probability of those 4 numbers, so that you do not count the probability of those numbers twice.

### Review

1. State the Addition Rule for probability and explain when it is used.

2. What happens to the Addition Rule when the two events considered are disjoint?

3. Use a Venn diagram to help explain why there is subtraction in the Addition Rule.

4. Sarah tells her mom that there is a 40% chance she will clean her room, a 70% she will do her homework, and a 24% chance she will clean her room and do her homework. What is the probability of Sarah cleaning her room or doing her homework?

5. You dad only ever makes one meal for dinner. The probability that he makes pizza tonight is 30%. The probability that he makes pasta tonight is 60%. What is the probability that he makes pizza or pasta?

6. After your little sister has gone trick-or-treating for Halloween, your mom says she is allowed to have 2 pieces of candy. The probability of her having a Snickers is 50%. The probability of her having a peanut butter cup is 60%. The probability of her having a Snickers or a peanut butter cup is 100%. What is the probability of her having a Snickers and a peanut butter cup?

7. Deanna sometimes likes honey or lemon in her tea. There is a 50% chance that she will have honey and lemon, a 95% chance that she will have honey or lemon, and a 80% chance that she will have honey. What is the probability that she will have lemon?

Consider the experiment of drawing a card from a standard deck. Let  be the event that the card is a diamond. Let  be the event that the card is a Jack. Let  be the event that the card is a four.

8. Find .

9. Find . What does this probability represent compared to ?

10. To find , all you need to do is add  and . Why is this and why do you not have to subtract anything?

Consider the experiment of flipping three coins and recording the sequence of heads and tails. Let  be the event that there is at least one heads. Let  be the event that the third coin is a tails. Let  be the event that the first coin is a heads.

11. Find . What does this probability represent?

12. Create a Venn diagram to show events  and  for this experiment.

13. Find  and . Compare and contrast these two probabilities.

The Addition Rule can be extended for three events. Consider three events that all share outcomes, as shown in the Venn diagram below.

14. Label the shaded part of the diagram in terms of .

15. Find a rule for  in terms of .

To see the Review answers, open this PDF file and look for section 11.6.

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

The Additive Rule of Probability states that the union of two events can be found by adding the probabilities of each event and subtracting the intersection of the two events

Intersection

Intersection is the probability of both or all of the events you are calculating happening at the same time (less likely).

probability

The chance that something will happen.

union

$\cup$ is a symbol that stands for union and is used to connect two groups together. It is associated with the logical term OR.