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Ch. 4 - Discrete Probability Distributions
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 4 - Discrete Probability DistributionsProblem 4.Q.2a
Chapter 4, Problem 4.Q.2a

The table lists the number of wireless devices per household in a small town in the United States.
Table showing the number of wireless devices per household and corresponding household counts in a small town.
a. Construct a probability distribution.

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Step 1: Calculate the total number of households by summing all the values in the 'Number of households' row. This will be the denominator for calculating probabilities.
Step 2: For each number of wireless devices (0, 1, 2, 3, 4, 5), divide the corresponding number of households by the total number of households to calculate the probability for each outcome. Use the formula: \( P(X = x) = \frac{\text{Number of households for } x}{\text{Total number of households}} \).
Step 3: Construct the probability distribution table. The table should have two columns: one for the number of wireless devices (0, 1, 2, 3, 4, 5) and another for the corresponding probabilities calculated in Step 2.
Step 4: Verify that the sum of all probabilities equals 1. This is a key property of a probability distribution and ensures correctness.
Step 5: Interpret the probability distribution. For example, identify which number of wireless devices is most likely (highest probability) and which is least likely (lowest probability).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Probability Distribution

A probability distribution describes how the probabilities are distributed over the values of a random variable. In this context, it represents the likelihood of each household having a specific number of wireless devices. The probabilities are calculated by dividing the number of households for each category by the total number of households.
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Relative Frequency

Relative frequency is the ratio of the number of times an event occurs to the total number of trials or observations. In constructing a probability distribution from the table, relative frequency is used to determine the probability of each number of wireless devices by taking the count of households for each device number and dividing it by the total number of households.
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Cumulative Frequency

Cumulative frequency is the sum of the frequencies for all categories up to a certain point. While not directly required for constructing a basic probability distribution, understanding cumulative frequency can help in analyzing the data further, such as determining the proportion of households with a certain number of devices or fewer.
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Related Practice
Textbook Question

In Exercises 21–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities

Thirty-six percent of Americans think there is still a need for the practice of changing their clocks for Daylight Savings Time. You randomly select seven Americans. Find the probability that the number who say there is still a need for changing their clocks for Daylight Savings Time is (c) at least six.

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Textbook Question

The five-year survival rate of people who undergo a liver transplant is 75%. The surgery is performed on six patients. (Source: Mayo Clinic)

b. Graph the binomial distribution using a histogram and describe its shape.

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Textbook Question

The five-year survival rate of people who undergo a liver transplant is 75%. The surgery is performed on six patients. (Source: Mayo Clinic)

a. Construct a binomial distribution.

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Textbook Question

In the past year, thirty-three percent of U.S. adults have put off medical treatment because of the cost. You randomly select nine U.S. adults. Find the probability that the number who have put off medical treatment because of the cost in the past year is (a) exactly three, (b) at most four, and (c) more than five. (Source: Gallup)

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Textbook Question

The table lists the number of wireless devices per household in a small town in the United States.

c. Find the mean, variance, and standard deviation of the probability distribution and interpret the results.

Textbook Question

In Exercises 1 and 2, determine whether the random variable x is discrete or continuous. Explain.


Let x represent the grade on an exam worth a total of 100 points.

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