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# 4.2: Probability Distribution for a Discrete Random Variable

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Know and understand the notion of discrete random variables.
• Learn how to use discrete random variables to solve probabilities of outcomes.

## Introduction

In this lesson, you will learn how to construct a probability distribution for a discrete random variable and represent this probability distribution with a graph, a table, or a formula. You will also learn the two conditions that all probability distributions must satisfy.

### Probability Distribution for a Discrete Random Variable

The example below illustrates how to specify the possible values that a discrete random variable can assume.

Example: Suppose you simultaneously toss two fair coins. Let X\begin{align*}X\end{align*} be the number of heads observed. Find the probability associated with each value of the random variable X\begin{align*}X\end{align*}.

Since there are two coins, and each coin can be either heads or tails, there are four possible outcomes (HH,HT,TH,TT)\begin{align*}(HH, HT, TH, TT)\end{align*}, each with a probability of 14\begin{align*}\frac{1}{4}\end{align*}. Since X\begin{align*}X\end{align*} is the number of heads observed, x=0,1,2.\begin{align*}x= 0, 1, 2.\end{align*}

We can identify the probabilities of the simple events associated with each value of X\begin{align*}X\end{align*} as follows:

P(x=0)P(x=1)P(x=2)=P(TT)=14=P(HT)+P(TH)=14+14=12=P(HH)=14\begin{align*}P(x=0) &= P(TT)=\frac{1}{4}\\ P(x=1) &= P(HT)+P(TH)=\frac{1}{4}+\frac{1}{4}=\frac{1}{2}\\ P(x=2) &= P(HH)=\frac{1}{4}\end{align*}

This is a complete description of all the possible values of the random variable, along with their associated probabilities. We refer to this as a probability distribution. This probability distribution can be represented in different ways. Sometimes it is represented in tabular form and sometimes in graphical form. Both forms are shown below.

In tabular form:

x\begin{align*}x\end{align*} P(x)\begin{align*}P(x)\end{align*}
0 14\begin{align*}\frac{1}{4}\end{align*}
1 12\begin{align*}\frac{1}{2}\end{align*}
2 14\begin{align*}\frac{1}{4}\end{align*}

Figure: The tabular form of the probability distribution for the random variable in the first example.

As a graph:

A probability distribution of a random variable specifies the values the random variable can assume, along with the probability of it assuming each of these values. All probability distributions must satisfy the following two conditions:

P(x)0,for all values of XP(x)=1,for all values of X\begin{align*}& P(x) \ge 0, \text{for all values of} \ X\\ & \sum_{} P(x)=1, \text{for all values of} \ X\end{align*}

Example: What is the probability distribution for the number of yes votes for three voters? (See the first example in the Chapter Introduction.)

Since each of the 8 outcomes is equally likely, the following table gives the probability of each value of the random variable.

Value of Random Variable (Number of Yes Votes) Probability
3 18=0.125\begin{align*}\frac{1}{8} = 0.125\end{align*}
2 38=0.375\begin{align*}\frac{3}{8}=0.375\end{align*}
1 38=0.375\begin{align*}\frac{3}{8}=0.375\end{align*}
0 18=0.125\begin{align*}\frac{1}{8} = 0.125\end{align*}

Figure: Tabular representation of the probability distribution for the random variable in the first example in the Chapter Introduction.

## Lesson Summary

The probability distribution of a discrete random variable is a graph, a table, or a formula that specifies the probability associated with each possible value that the random variable can assume.

All probability distributions must satisfy the following two conditions:

P(x0),for all values of XP(x)=1,for all values of X\begin{align*}& P(x \ge 0), \text{for all values of} \ X\\ & \sum_{} P(x)=1, \text{for all values of} \ X\end{align*}

## Review Questions

1. Consider the following probability distribution: xP(x)40.100.310.430.2\begin{align*}& x && -4 && 0 && 1 && 3\\ & P(x) && 0.1 && 0.3 && 0.4 && 0.2\end{align*}
1. What are all the possible values of X\begin{align*}X\end{align*}?
2. What value of X\begin{align*}X\end{align*} is most likely to happen?
3. What is the probability that x>0\begin{align*}x > 0\end{align*}?
4. What is the probability that x=2\begin{align*}x = -2\end{align*}?
2. A fair die is tossed twice, and the up face is recorded each time. Let X\begin{align*}X\end{align*} be the sum of the up faces.
1. Give the probability distribution for X\begin{align*}X\end{align*} in tabular form.
2. What is P(x8)\begin{align*}P(x \ge 8)\end{align*}?
3. What is P(x<8)\begin{align*}P(x < 8)\end{align*}?
4. What is the probability that x\begin{align*}x\end{align*} is odd? What is the probability that x\begin{align*}x\end{align*} is even?
5. What is P(x=7)\begin{align*}P(x=7)\end{align*}?
3. If a couple has three children, what is the probability that they have at least one boy?

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