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.

검증된 단계별 안내

Step 1: Identify the problem type. This is a binomial probability problem because there are a fixed number of trials (n = 9), two possible outcomes (having access to medical care benefits or not), and a constant probability of success (p = 0.72).

Step 2: Define the random variable X. Let X represent the number of employees (out of 9) who have access to medical care benefits. X follows a binomial distribution: X ~ Binomial(n = 9, p = 0.72).

Step 3: Translate the problem into a probability statement. The problem asks for the probability that more than six employees have access to medical care benefits. This can be written as P(X > 6).

Step 4: Use the complement rule to simplify the calculation. P(X > 6) can be rewritten as 1 - P(X ≤ 6). This means you need to calculate the cumulative probability P(X ≤ 6) and subtract it from 1.

Step 5: Use the binomial probability formula or technology to calculate P(X ≤ 6). The binomial probability formula is P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where 'n choose k' is the binomial coefficient. Sum the probabilities for X = 0, 1, 2, ..., 6 to find P(X ≤ 6), then subtract this value from 1 to get P(X > 6). Alternatively, use a statistical calculator or software to compute P(X > 6) directly.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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, 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).

Probability Calculation

To find the probability of a specific 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 different outcomes.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a certain threshold. In this case, to find the probability that more than six employees have access to medical care benefits, we can calculate the cumulative probability for six or fewer employees and subtract it from one. This approach simplifies the calculation by leveraging previously computed probabilities.