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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.RE.12
Chapter 4, Problem 4.RE.12

In Exercises 11 and 12, determine whether the experiment is a binomial experiment. If it is, identify a success; specify the values of n, p, and q; and list the possible values of the random variable x. If it is not a binomial experiment, explain why.


A fair coin is tossed repeatedly until 15 heads are obtained. The random variable x counts the number of tosses.

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Step 1: Recall the criteria for a binomial experiment. A binomial experiment must satisfy the following conditions: (1) The experiment consists of a fixed number of trials, n. (2) Each trial has only two possible outcomes: success or failure. (3) The probability of success, p, is the same for each trial. (4) The trials are independent of each other.
Step 2: Analyze the given problem. Here, a fair coin is tossed repeatedly until 15 heads are obtained. The random variable x counts the number of tosses. Note that the number of trials (tosses) is not fixed in advance; instead, the experiment continues until a specific number of successes (15 heads) is achieved.
Step 3: Determine whether the problem satisfies the binomial criteria. Since the number of trials is not fixed and depends on the outcome (stopping after 15 heads), this violates the first condition of a binomial experiment. Therefore, this is not a binomial experiment.
Step 4: Explain why this is not a binomial experiment. The key reason is that the number of trials is not predetermined. Instead, the experiment is based on achieving a fixed number of successes (15 heads), which aligns more with a negative binomial distribution rather than a binomial distribution.
Step 5: Conclude the analysis. Since the problem does not meet the criteria for a binomial experiment, it cannot be analyzed using binomial parameters such as n, p, q, or the possible values of x. Instead, it would require a different statistical approach, such as using the negative binomial distribution.

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Key Concepts

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

Binomial Experiment

A binomial experiment consists of a fixed number of independent trials, each with two possible outcomes: success or failure. The probability of success, denoted as p, remains constant across trials. The random variable x represents the number of successes in these trials. For an experiment to be classified as binomial, it must meet these criteria, including a defined number of trials.
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Random Variable

A random variable is a numerical outcome of a random process. In the context of a binomial experiment, it typically counts the number of successes in a series of trials. The possible values of the random variable x can range from 0 to n, where n is the total number of trials. Understanding the nature of the random variable is crucial for analyzing the distribution and probabilities associated with the experiment.
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Geometric Distribution

The geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. Unlike a binomial experiment, it does not have a fixed number of trials; instead, it continues until a specified outcome occurs. In the given scenario, since the coin is tossed until 15 heads are obtained, it aligns with a geometric distribution rather than a binomial one, as the number of trials is not predetermined.
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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.

Fourteen percent of noninstitutionalized U.S. adults smoke cigarettes. After randomly selecting ten noninstitutionalized U.S. adults, you ask them whether they smoke cigarettes. Find the probability that the first adult who smokes cigarettes is (b) the fourth or fifth person selected.

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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.

Fourteen percent of noninstitutionalized U.S. adults smoke cigarettes. After randomly selecting ten noninstitutionalized U.S. adults, you ask them whether they smoke cigarettes. Find the probability that the first adult who smokes cigarettes is (a) the third person selected.

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Textbook Question

In Exercises 13–16, find the indicated binomial probabilities. If convenient, use technology or Table 2 in Appendix B.

Seventy-two percent of U.S. civilian employees have access to medical care benefits. You randomly select nine civilian employees. Find the probability that the number who have access to medical care benefits is (b) at least six

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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 (a) exactly four

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Textbook Question

In Exercises 13–16, find the indicated binomial probabilities. If convenient, use technology or Table 2 in Appendix B.

Fifty-three percent of U.S. adults support attempting to land an astronaut on Mars. You randomly select eight U.S. adults. Find the probability that the number who support attempting to land an astronaut on Mars is (b) at least three

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Textbook Question

In Exercises 13–16, find the indicated binomial probabilities. If convenient, use technology or Table 2 in Appendix B.

Seventy-two percent of U.S. civilian employees have access to medical care benefits. You randomly select nine civilian employees. Find the probability that the number who have access to medical care benefits is (c) more than six.

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