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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.15a
Chapter 4, Problem 4.RE.15a

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 (a) exactly six

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1
Step 1: Recognize that this is a binomial probability problem. The binomial distribution is used when there are a fixed number of independent trials, each with two possible outcomes (success or failure). Here, the number of trials (n) is 9, the probability of success (p) is 0.72, and the number of successes (x) is 6.
Step 2: Write the formula for the binomial probability: P(X = x) = (n choose x) * p^x * (1 - p)^(n - x). Here, (n choose x) is the binomial coefficient, which can be calculated as n! / [x! * (n - x)!].
Step 3: Substitute the given values into the formula. For this problem, n = 9, x = 6, and p = 0.72. The formula becomes: P(X = 6) = (9 choose 6) * (0.72)^6 * (1 - 0.72)^(9 - 6).
Step 4: Calculate the binomial coefficient (9 choose 6). This is done using the formula: (9 choose 6) = 9! / [6! * (9 - 6)!]. Simplify this expression to find the value of the coefficient.
Step 5: Compute the probability by multiplying the binomial coefficient by (0.72)^6 and (1 - 0.72)^3. This will give you the probability that exactly 6 out of 9 employees have access to medical care benefits.

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Key Concepts

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

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, it applies to the scenario of selecting employees with a specific probability of having medical care benefits. The distribution is defined by two parameters: the number of trials (n) and the probability of success (p).
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Probability Mass Function (PMF)

The probability mass function for a binomial distribution gives the probability of obtaining exactly k successes in n trials. It is calculated using the formula P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n choose k' represents the binomial coefficient. This function is essential for determining the likelihood of specific outcomes, such as exactly six employees having access to benefits.
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Binomial Coefficient

The binomial coefficient, denoted as 'n choose k' or C(n, k), represents the number of ways to choose k successes from n trials. It is calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. This concept is crucial for calculating probabilities in binomial distributions, as it quantifies the different combinations of successes and failures.
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Related Practice
Textbook Question

The Centers for Disease Control and Prevention (CDC) is required by law to publish a report on assisted reproductive technology (ART). ART includes all fertility treatments in which both the egg and the sperm are used. These procedures generally involve removing eggs from a patient’s ovaries, combining them with sperm in the laboratory, and returning them to the patient’s body or giving them to another patient.

You are helping to prepare a CDC report on young ART patients and select at random 6 ART cycles of patients under 35 years of age for a special review. None of the cycles resulted in a live birth. Your manager feels it is impossible to select at random 10 ART cycles that do not result in a live birth. Use the pie chart at the right and your knowledge of statistics to determine whether your manager is correct.

a. How would you determine whether your manager is correct, that it is impossible to select at random six ART cycles that do not result in a live birth?

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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

89
views
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

105
views
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.

74
views
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 (a) exactly three

128
views
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 (c) more than three.

85
views