# 3.1: What are Variables?

**Learning Objectives**

- Demonstrate an understanding of the notion of discrete random variables by using them to solve for the probabilities of outcomes, such as the probability of the occurrence of 5 heads in 14 coin tosses.

In the previous 2 chapters, we looked at the mathematics involved in probability events. We looked at examples of event occurring if event had occurred (conditional events), of event being affected by the outcome of event (dependent events), and of event and event not being affected by each other (independent events). We also looked at examples where events cannot occur at the same time (mutually exclusive events), or when events were not mutually exclusive and there was some overlap, so that we had to account for the double counting (mutually inclusive events). If you recall, we used Venn Diagrams (below), tree diagrams, and even tables to help organize information in order to simplify the mathematics for the probability calculations.

Our examination of probability, however, began with a look at the English language. Although there are a number of differences in what terms mean in mathematics and English, there are a lot of similarities as well. We saw this with the terms independent and dependent. In this chapter, we are going to learn about variables. In particular, we are going to look at discrete random variables. When you see the sequence of words *discrete random variables*, it may, at first, send a shiver down your spine, but let’s look at the words individually and see if we can "simplify" the sequence!

The term discrete, in English, means to constitute a separate thing or to be related to unconnected parts. In mathematics, we use the term discrete when we are talking about pieces of data that are not connected. Random, in English, means to lack any plan or to be without any prearranged order. In mathematics, the definition is the same. Random events are fair, meaning that there is no way to tell what outcome will occur. In the English language, the term variable means to be likely to change or subject to variation. In mathematics, the term variable means to have no fixed quantitative value.

Now that we have seen the 3 terms separately, let’s combine them and see if we can come up with a definition of a discrete random variable. We can say that discrete variables have values that are unconnected to each other and have variations within the values. Think about the last time you went to the mall. Suppose you were walking through the parking lot and were recording how many cars were made by Ford. The variable is the number of Ford cars you see. Therefore, since each car is either a Ford or it is not, the variable is discrete. Also, **random variables** are simply quantities that take on different values depending on chance, or probability. Thus, if you randomly selected 20 cars from the parking lot and determined whether or not each was manufactured by Ford, you would then have a discrete random variable.

Now let’s define discrete random variables. **Discrete random variables** represent the number of distinct values that can be counted of an event. For example, when Robert was randomly chosen from all the students in his classroom and asked how many siblings there are in his family, he said that he has 6 sisters. Joanne picked a random bag of jelly beans at the store, and only 15 of 250 jelly beans were green. When randomly selecting from the most popular movies, Jillian found that *Iron Man 2* grossed 3.5 million dollars in sales on its opening weekend. Jack, walking with his mom through the parking lot, randomly selected 10 cars on his way up to the mall entrance and found that only 2 were Ford vehicles.