# Chapter 3: Introduction to Discrete Random Variables

**Basic**Created by: CK-12

## Introduction

In this chapter, you will learn about discrete random variables. Discrete random variables can take on a finite number of values in an interval, or as many values as there are positive integers. In other words, a discrete random variable can take on an infinite number of values, but not all the values in an interval. When you find the probabilities of these values, you are able to show the probability distribution. A probability distribution consists of all the values of the random variable, along with the probability of the variable taking on each of these values. Each probability must be between 0 and 1, and the probabilities must sum to 1.

You will also be introduced to the concept of a binomial distribution. This will be discussed in depth in the next chapter, but in this chapter, you will use a binomial distribution when talking about the number of successful events or the value of a random variable. A binomial distribution is only used when there are 2 possible outcomes. For example, you will use the binomial distribution formula for coin tosses (heads or tails). Other examples include yes/no responses, true or false questions, and voting Democrat or Republican. When the number of possible outcomes goes beyond 2, you use a multinomial distribution. Rolling a die is a common example of a multinomial distribution problem.

In addition, you will use factorials again for solving these problems. Factorials were introduced in Chapter 2 for permutations and combinations, but they are also used in many other probability problems. Finally, you will use a graphing calculator to show the difference between theoretical and experimental probability. The calculator is an effective and efficient tool for illustrating the difference between these 2 probabilities, and also for determining the experimental probability when the number of trials is large.

- 3.1.
## Discrete Random Variables

- 3.2.
## Probability Distribution

- 3.3.
## Binomial Distributions and Probability

- 3.4.
## Multinomial Distributions

- 3.5.
## Theoretical and Experimental Spinners

- 3.6.
## Theoretical and Experimental Coin Tosses

### Chapter Summary

## Summary

This chapter covers discrete random variables and probability distributions. Discrete random variables represent the number of distinct values that can be counted of an event. *Discrete* means distinct values like a number of cards as opposed to continuous distribution like amount of water. A probability distribution is a table, a graph, or a chart that shows you all the possible values of a discrete random variable and the probabilities of each. Binomial distributions are a particular case of getting \begin{align*}X\end{align*} successes in \begin{align*}n\end{align*} trials. If \begin{align*}a\end{align*} is the number of successes, \begin{align*}p\end{align*} is the probability of the event occurring, and \begin{align*}q\end{align*} is the probability of the event not occurring, the binomial probability is:

\begin{align*}P(X = a) = {_n}C_a \times p^a \times q^{(n - a)}\end{align*}

Multinomial distributions are a further case where there are outcomes beyond success and failure. If \begin{align*}n\end{align*} is the number of trials, \begin{align*}p\end{align*} is the probability for each possible outcome, and \begin{align*}k\end{align*} is the number of possible outcomes, the multinomial probability is:

\begin{align*}P & = \frac{n!}{n_1!n_2!n_3!\ldots n_k!} \times \left (p_1{^{n_1}} \times p_2{^{n_2}} \times p_3{^{n_3}} \ldots p_k{^{n_k}} \right )\end{align*}

The chapter concludes with notes on using a graphing calculator when solving these problems.