**The Addition Rule for Probability**

The **addition rule for probability** says the probability that event A *or *event B occurs is equal to the probability that event A occurs plus the probability that event B occurs minus the probability that event A *and* event B both occur.

We denote “the probability that event A *or *event B occurs” as P(A ∪ B) and “the probability that event A *and* event B both occur” as P(A ∩ B).

Thus, the probability that even A *or *event B occurs is written as:

P(A ∪ B) = P(A) + P(B) – P(A ∩ B)

**Example 1: Bob goes to the grocery. The probability that he buys an apple is 0.5, milk is 0.3, and both an apple and milk is 0.1. What is the probability that he buys an apple or milk?**

P(Apple ∪ Milk) = P(Apple) + P(Milk) – P(Apple ∩ Milk)

P(Apple ∪ Milk) = 0.5 + 0.3 – 0.1 = **0.7**

**Mutually Exclusive Events**

Two events are **mutually exclusive** if the probability that they *both *occur is equal to 0.

If events A and B are mutually exclusive, then the probability that event A *or *B occurs is equal to:

P(A ∪ B) = P(A) + P(B)

**Example 2: A bag has 3 blue marbles, 5 red marbles, and 2 purple marbles. If you randomly grab one marble out of the bag, what is the probability that you select a blue or red marble?**

P(red ∪ blue) = P(red) + P(blue) – P(red ∩ blue)

P(red ∪ blue) = 0.5 + 0.3 – 0 = **0.8**

In this case, it’s not possible to choose *both *a red and blue marble in one single grab, so these events are **mutually exclusive**.

This means we could simplify the equation above to:

P(red ∪ blue) = P(red) + P(blue)

P(red ∪ blue) = 0.5 + 0.3 = **0.8**

**Complete Example**

**Example 3: 400 students were asked whether they prefer pizza or ice cream. The two-way table below shows the results:**

**What is the probability that a randomly selected student prefers pizza?**

The probability that a randomly selected student prefers pizza is 0.4, or 40%.

**What is the probability that a randomly selected student prefers pizza or prefers ice cream?**

P(prefers pizza or prefers ice cream) = P(prefers pizza) + P(prefers ice cream) – P(prefers pizza *and *ice cream)

P(prefers pizza or prefers ice cream) = (160/400) + (150/400) – (0/100)

P(prefers pizza or prefers ice cream) = (0.4) + (0.375) – (0) = **0.775**

**In this sample, are the events “prefers pizza” and “prefers ice cream” mutually exclusive?**

Notice that there is no option for a student to say they prefer pizza *and *ice cream. They’re forced to choose pizza, ice cream, or no preference. So in this case, the events “prefers pizza” and “prefers ice cream” are mutually exclusive.

**What is the probability that a randomly selected student prefers pizza or is male?**

P(prefers pizza or is male) = P(prefers pizza) + P(is male) – P(prefers pizza *and *is male)

P(prefers pizza or is male) = (160/400) + (250/400) – (100/400)

P(prefers pizza or is male) = (0.4) + (0.625) – (0.25) = **0.775**

**In this sample, are the events “prefers pizza” and “is male” mutually exclusive?**

Notice that it’s possible for a student to prefer pizza *and *be male. This means these two events are not mutually exclusive.