# Probability

## The Addition Rule for Probability


### Learning Objectives

After completing this section, you should be able to:
1. Identify mutually exclusive events.
2. Apply the Addition Rule to compute probability.
3. Use the Inclusion/Exclusion Principle to compute probability.
Up to this point, we have looked at the probabilities of simple events. Simple events are those with a single, simple characterization. Sometimes, though, we want to investigate more complicated situations. For example, if we are choosing a college student at random, we might want to find the probability that the chosen student is a varsity athlete or in a Greek organization. This is a compound event: there are two possible criteria that might be met. We might instead try to identify the probability that the chosen student is both a varsity athlete and in a Greek organization. In this section and the next, we’ll cover probabilities of two types of compound events: those build using “or” and those built using “and.” We’ll deal with the former first.

### Mutual Exclusivity

Before we get to the key techniques of this section, we must first introduce some new terminology. Let’s say you’re drawing a card from a standard deck. We’ll consider 3 events:  is the event “the card is a ,”  is the event “the card is a 10,” and  is the event “the card is a .” If the card drawn is , then  and  didn’t occur, but  did. If the card drawn is instead , then  didn’t occur, but both  and  did.
We can see from these examples that, if we are interested in several possible events, more than one of them can occur simultaneously (both  and , for example). But, if you think about all the possible outcomes, you can see that  and  can never occur simultaneously; there are no cards in the deck that are both  and . Pairs of events that cannot both occur simultaneously are called mutually exclusive. Let’s go through an example to help us better understand this concept.

### The Addition Rule for Mutually Exclusive Events

If two events are mutually exclusive, then we can use addition to find the probability that one or the other event occurs.
Why does this formula work? Let’s consider a basic example. Suppose we’re about to draw a Scrabble tile from a bag containing A, A, B, E, E, E, R, S, S, U. What is the probability of drawing an E or an S? Since 3 of the tiles are marked with E and 2 are marked with S, there are 5 tiles that satisfy the criteria. There are ten tiles in the bag, so the probability is . Notice that the probability of drawing an E is  and the probability of drawing an S is ; adding those together, we get . Look at the numerators in the fractions involved in the sum: the 3 represents the number of E tiles and the 2 is the number of S tiles. This is why the Addition Rule works: The total number of outcomes in one event or the other is the sum of the numbers of outcomes in each of the individual events.

### Finding Probabilities When Events Aren’t Mutually Exclusive

Let’s return to the example we used to explore the Addition Rule: We’re about to draw a Scrabble tile from a bag containing A, A, B, E, E, E, R, S, S, U. Consider these events:  is “draw a vowel” and  is “draw a letter that comes after L in the alphabet.” Since there are 6 vowels, . There are 4 tiles with letters that come after L alphabetically, so . What is ? If we blindly apply the Addition Rule, we get , which would mean that the compound event  or  is certain. However, it’s possible to draw a B, in which case neither  nor  happens. Where’s the error?
The events are not mutually exclusive: the outcome U belongs to both events, and so the Addition Rule doesn’t apply. However, there’s a way to extend the Addition Rule to allow us to find this probability anyway; it’s called the Inclusion/Exclusion Principle. In this example, if we just add the two probabilities together, the outcome U is included in the sum twice: It’s one of the 6 outcomes represented in the numerator of , and it’s one of the 4 outcomes represented in the numerator of . So, that particular outcome has been “double counted.” Since it has been included twice, we can get a true accounting by excluding it once: . We can generalize this idea to a formula that we can apply to find the probability of any compound event built using “or.”
It’s worth noting that this formula is truly an extension of the Addition Rule. Remember that the Addition Rule requires that the events  and  are mutually exclusive. In that case, the compound event  is impossible, and so . So, in cases where the events in question are mutually exclusive, the Inclusion/Exclusion Principle reduces to the Addition Rule.

### Check Your Understanding


### Key Terms

1. mutually exclusive

### Key Concepts

1. The Addition Rule is used to find the probability that one event or another will occur when those events are mutually exclusive.
2. The Inclusion/Exclusion Principle is used to find probabilities when events are not mutually exclusive.

### Formulas

1. If  and  are mutually exclusive events, then
    
    
2. If  and  are events that contain outcomes of a single experiment, then
    
    