### Discrete Random Variables

In previous Concepts, we looked at the mathematics involved in probability events. We looked at examples of event \begin{align*}A\end{align*} occurring if event \begin{align*}B\end{align*} had occurred (conditional events), of event \begin{align*}B\end{align*} being affected by the outcome of event \begin{align*}A\end{align*} (dependent events), and of event \begin{align*}A\end{align*} and event \begin{align*}B\end{align*} 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 and the following Concepts, 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.

The probability of a discrete random variable can range anywhere from 0 to 1. The less likely a discrete random variable is to occur, the closer the probability will be to 0, and the more likely a discrete random variable is to occur, the closer the probability will be to 1.

#### Using Discrete Random Variables

Which of the following can be represented by a discrete random variable?

a. The heights of the students in a high school

Cannot be represented by a discrete random variable.

b. The number of sit-ups that you can do

Can be represented by a discrete random variable.

c. The distances between stars in a galaxy

Cannot be represented by a discrete random variable.

d. The number of wins by a professional hockey team

Can be represented by a discrete random variable.

e. The speeds of the cars in a race

Cannot be represented by a discrete random variable.

The heights of the students in a high school cannot be represented by a discrete random variable, since a height can take on any value within a certain range. For example, a height could be 64 inches, 64.5 inches, 64.55 inches, 64.555 inches, and so on. On the other hand, the number of sit-ups that you can do can be represented by a discrete random variable, because the number will always be an integer. The distances between stars in a galaxy are similar to the heights of the students in a high school in that there are an infinite number of possibilities, so these distances cannot be represented by a discrete random variable. However, the number of wins by a professional hockey team is similar to the number of sit-ups that you can do in that the number will always be an integer, so this number can be represented by a discrete random variable. Finally, the speeds of the cars in a race can take on any values, such as 100 MPH, 100.1 MPH, 100.12 MPH, 100.123 MPH, and so on, so these speeds cannot be represented by a discrete random variable. In summary, the answers to this question are as follows:

#### Determining Probability

It is very likely, but not certain, that the high temperature will exceed \begin{align*}75^\circ \text{F}\end{align*} every day next week. Suppose that the discrete random variable \begin{align*}X\end{align*} represents the number of days next week that the high temperature will exceed \begin{align*}75^\circ \text{F}\end{align*}. Which of these could be \begin{align*}P(X=7)\end{align*}?

a. \begin{align*}P(X=7)=0\end{align*}

b. \begin{align*}P(X=7)=0.14\end{align*}

c. \begin{align*}P(X=7)=0.5\end{align*}

d. \begin{align*}P(X=7)=0.96\end{align*}

e. \begin{align*}P(X=7)=1\end{align*}

If the probability of an event is 0, it is impossible, and if the probability of the event is 1, it is certain. In this case, it is very likely, but not certain, that the high temperature will exceed \begin{align*}75^\circ \text{F}\end{align*} every day next week, so the probability will be close to 1, but not 1. Therefore, \begin{align*}P(X=7)\end{align*} could be equal to 0.96, so the correct answer is d.

#### Understanding Probability

Suppose that the discrete random variable \begin{align*}X\end{align*} represents the number of points out of 100 that Royce scores on a test. If \begin{align*}P(X=42)=0.88\end{align*}, which of these statements is most likely true?

a. Royce is likely to do well on the test.

b. Royce doesn't even know a single correct answer on the test.

c. Royce didn't study much for the test.

d. Royce knows all the correct answers on the test.

e. Royce studied a lot for the test.

Since \begin{align*}P(X=42)=0.88\end{align*}, the probability is high that Royce will get a low score on the test. However, if Royce gets 42 points out of 100, he at least knows some correct answers on the test. Therefore, the statement that is most likely true is that Royce didn't study much for the test, so the correct answer is C.

### Examples

Put the following statements in order from least likely to most likely.

#### Example 1

If a die is rolled 2 times, the same number will come up each time.

Probability = \begin{align*}\frac{1}{6}\end{align*}.

#### Example 2

Christmas will be in December next year.

Likelihood is extremely high.

#### Example 3

A letter chosen at random from the alphabet will be a consonant.

Probability = \begin{align*}\frac{21}{26}\end{align*}.

#### Example 4

If a couple has a baby, it will be a girl.

Probability is about \begin{align*}\frac{1}{2}\end{align*}.

#### Example 5

The population of Wyoming will be greater than that of California in 10 years.

Probability is extremely low.

We first looked at each statement individually. The first statement is, "If a die is rolled 2 times, the same number will come up each time." The probability of this statement can actually be calculated to be \begin{align*}\frac{1}{6}\end{align*}. Next, we have, "Christmas will be in December next year." This probability of this statement is extremely high, as Christmas has traditionally been in December every year. After this, we have, "A letter chosen at random from the alphabet will be a consonant." Since 21 out of 26 letters in the alphabet are consonants, the probability of this statement is \begin{align*}\frac{21}{26}\end{align*}. The next statement is, "If a couple has a baby, it will be a girl." The probability of this statement will be about \begin{align*}\frac{1}{2}\end{align*}, since a boy and a girl are almost equally likely. Finally, we have, "The population of Wyoming will be greater than that of California in 10 years." The probability of this statement is extremely low, as the population of California is currently about 37,000,000, while the population of Wyoming is less than 600,000. In summary, here are our probabilities:c. \begin{align*}\frac{21}{26}\end{align*}

Therefore, the order of the statements from least likely to most likely is e, a, d, c, b.

### Review

- Match the following statements from the first column with the probability values in the second column.

Probability Statement |
\begin{align*}P(X)\end{align*} |
---|---|

a. The probability of this event will never occur. | \begin{align*}\underline{\;\;\;\;\;} \ P(X) = 1.0\end{align*} |

b. The probability of this event is highly likely. | \begin{align*}\underline{\;\;\;\;\;} \ P(X) = 0.33\end{align*} |

c. The probability of this event is very likely. | \begin{align*}\underline{\;\;\;\;\;} \ P(X) = 0.67\end{align*} |

d. The probability of this event is somewhat likely. | \begin{align*}\underline{\;\;\;\;\;} \ P(X) = 0.00\end{align*} |

e. The probability of this event is certain. | \begin{align*}\underline{\;\;\;\;\;} \ P(X) = 0.95\end{align*} |

- Match the following statements from the first column with the probability values in the second column.

Probability Statement |
\begin{align*}P(X)\end{align*} |
---|---|

a. I bought a ticket for the State Lottery. The probability of a successful event (winning) is likely to be: | \begin{align*}\underline{\;\;\;\;\;} \ P(X) = 0.80\end{align*} |

b. I have a bag of equal numbers of red and green jelly beans. The probability of reaching into the bag and picking out a red jelly bean is likely to be: | \begin{align*}\underline{\;\;\;\;\;} \ P(X) = 0.50\end{align*} |

c. My dad teaches math, and my mom teaches chemistry. The probability that I will be expected to study science or math is likely to be: | \begin{align*}\underline{\;\;\;\;\;} \ P(X) = 0.67\end{align*} |

d. Our class has the highest test scores in the State Math Exams. The probability that I have scored a great mark is likely to be: | \begin{align*}\underline{\;\;\;\;\;} \ P(X) = 1.0\end{align*} |

e. The Chicago baseball team has won every game this season. The probability that the team will make it to the playoffs is likely to be: | \begin{align*}\underline{\;\;\;\;\;} \ P(X) = 0.01\end{align*} |

- Read each of the following statements and match the following words to each statement. You can put your answers directly into the table. Here is the list of terms you can add:
- certain or sure
- impossible
- likely or probable
- unlikely or improbable
- maybe
- uncertain or unsure

Statement |
Probability Term |
---|---|

Tomorrow is Friday. | |

I will be in New York on Friday. | |

It will be dark tonight. | |

It is snowing in August! | |

China is cold in January. |

- Read each of the following statements and match the following words to each statement. You can put your answers directly into the table. Here is the list of terms you can add:
- certain or sure
- impossible
- likely or probable
- unlikely or improbable
- maybe
- uncertain or unsure

Statement |
Probability Term |
---|---|

I am having a sandwich for lunch. | |

I have school tomorrow. | |

I will go to the movies tonight. | |

January is warm in New York. | |

My dog will bark. |

- Can the amount of lemonade in a pitcher be represented by a discrete random variable? Why or why not?
- Can the number of musicians in an orchestra be represented by a discrete random variable? Why or why not?
- Can the weights of the alligators in a swamp be represented by a discrete random variable? Why or why not?
- Can the number of tickets sold to a movie be represented by a discrete random variable? Why or why not?
- Give an example of something that can be represented with a discrete random variable. Explain your answer.
- Give an example of something that cannot be represented with a discrete random variable. Explain your answer.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 3.1.