In this Concept, 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.

### Watch This

For an introduction to discrete probability distributions **(3.0)**, see statslectures, Discrete Probability Distributions (1:46).

### Guidance

**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 A

Suppose you simultaneously toss two fair coins. Let be the number of heads observed. Find the probability associated with each value of the random variable .

Since there are two coins, and each coin can be either heads or tails, there are four possible outcomes , each with a probability of . Since is the number of heads observed,

We can identify the probabilities of the simple events associated with each value of as follows:

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:

0 | |

1 | |

2 |

**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:

#### Example B

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. The value of the random variable is the *number of yes votes*.

Value of Random Variable |
Probability |
---|---|

3 | |

2 | |

1 | |

0 |

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

#### Example C

Consider the following two probability distributions:

X | 2 | 3 | 4 |

P(X) | 1/5 | 1/5 | 3/5 |

Y | 3 | 4 | 5 |

P(Y) | 3/5 | 1/5 | 1/5 |

The observed values are 3 and 4. Which observed value goes with which random variable? Are you sure? Explain.

**Solution:**

The observed value 3 is more likely to come from the second probability distribution. In this distribution the probability of obtaining a 3 is 3/5. In the first distribution the probability of obtaining a 3 is 1/5. The value 4 has a probability of 3/5 in the first distribution but only a probability of 1/5 in the second distribution. However, you can not be certain which distribution these values comes from.

### Guided Practice

Consider the following two probability distributions:

X | 1 | 2 | 3 |

P(X) | 1/3 | 1/3 | 1/3 |

Y | 4 | 5 | |

P(Y) | 1/3 | 2/3 |

There are two observed values: 3 and 4. Which observed value goes with which random variable? Are you sure? Explain.

**Solution:**

The 3 is a possible value of the random variable X. The random variable Y does not take on the value of 3. The value 4 is a possible value of the random variable Y. The random variable X does not take on the value 4.

### Explore More

- Consider the following probability distribution:
- What are all the possible values of ?
- What value of is most likely to happen?
- What is the probability that ?
- What is the probability that ?

- A fair die is tossed twice, and the up face is recorded each time. Let be the sum of the up faces.
- Give the probability distribution for in tabular form.
- What is ?
- What is ?
- What is the probability that is odd? What is the probability that is even?
- What is ?

- If a couple has three children, what is the probability that they have at least one boy?

- Suppose there are six numbers in a box: 1, 2, 3, 4, 5, 6.
- Suppose you draw two numbers with replacement. Are the draws independent? Explain.
- Suppose you draw two numbers without replacement. Are the draws independent? Explain.

- Two draws are made at random without replacement from a box with four numbers: 1,2, 3, 4. Find the probability that the second draw will be a 3. Explain.
- Suppose there is a box with four slips of paper each paper with one number: 1, 2, 3, 3. Let the random variable be defined as the number you choose at random. What is ?
- Suppose a box has four slips of paper and on each slip are two numbers. The slips of paper look like the following:
- Explain in words, what means.
- Find .
- Find .

- True or False? If and are independent then and are independent.
- Suppose two draws will be made at random with replacement from a box that has three slips of paper, each with a number on it: 1, 2, and 3. Let represent the first draw and represent the second draw.
- What is ?
- Find the chance that the first draw will be a one and the second draw will be a 2.
- Find
- Are and independent? Explain.
- Now suppose two draws are made at random without replacement. Are the variables independent? Explain.

- Suppose the random variable can take on the values 1 and 2 and the random variable can take on the values 1 and 3. If you are to be paid whatever value the random variable turns out to be, in dollars, which random variable do you prefer? Explain.
- Suppose for is a discrete probability distribution.
- Find
- Find

*Keywords*

Probability distribution

Random variables

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 4.2.