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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.15b
Chapter 4, Problem 4.RE.15b

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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Step 1: Identify the problem as 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 probability of failure (q) is 1 - p = 0.28.
Step 2: Define the random variable X as the number of employees who have access to medical care benefits. The problem asks for the probability that at least six employees have access to medical care benefits, which can be written as P(X ≥ 6).
Step 3: Use the complement rule to simplify the calculation. P(X ≥ 6) can be rewritten as 1 - P(X ≤ 5). This allows us to calculate the cumulative probability for X ≤ 5 and subtract it from 1.
Step 4: Use the binomial probability formula to calculate P(X ≤ 5). The formula is P(X = k) = (n choose k) * p^k * q^(n-k), where (n choose k) = n! / [k! * (n-k)!]. Compute the probabilities for X = 0, 1, 2, 3, 4, and 5, then sum them to find P(X ≤ 5).
Step 5: If convenient, use technology (e.g., a statistical calculator or software) to compute the cumulative probability P(X ≤ 5) directly. Subtract this value from 1 to find P(X ≥ 6).

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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, the trials are the selection of civilian employees, and a 'success' is defined as an employee having access to medical care benefits. The distribution is characterized by two parameters: the number of trials (n) and the probability of success (p).
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Probability Calculation

To find the probability of a certain number of successes in a binomial distribution, we use the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k). Here, 'n choose k' represents the number of ways to choose k successes from n trials, p is the probability of success, and (1-p) is the probability of failure. This formula allows us to calculate the likelihood of observing a specific number of employees with benefits.
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Cumulative Probability

Cumulative probability refers to the probability of obtaining a value less than or equal to a certain number in a distribution. In this case, to find the probability that at least six employees have access to medical care benefits, we can calculate the cumulative probabilities for six, seven, eight, and nine employees and sum these values. This approach is essential for determining probabilities for ranges of outcomes in binomial scenarios.
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Related Practice
Textbook Question

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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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

1
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.

1
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

1
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 (a) exactly six

1
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.

1
views