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 (c) not one of the first six persons selected.

검증된 단계별 안내

Step 1: Recognize that this problem involves a geometric distribution because we are looking for the probability that the first success (a smoker) occurs after a certain number of trials (the first six persons). The geometric distribution models the number of trials until the first success in a sequence of independent Bernoulli trials.

Step 2: Identify the probability of success (p) and failure (q). Here, the probability of success (a person smokes) is p = 0.14, and the probability of failure (a person does not smoke) is q = 1 - p = 0.86.

Step 3: Use the formula for the geometric distribution to calculate the probability that the first success occurs after the first six trials. The formula is P(X > k) = q^k, where k is the number of trials before the first success. In this case, k = 6.

Step 4: Substitute the values into the formula. Using q = 0.86 and k = 6, calculate P(X > 6) = (0.86)^6. This represents the probability that none of the first six persons selected are smokers.

Step 5: Interpret the result. Once the probability is calculated, compare it to a threshold (e.g., 0.05) to determine if the event is unusual. An event is typically considered unusual if its probability is less than 0.05.

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주요 개념

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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, where each trial has the same probability of success. In this context, it helps determine the probability that the first adult who smokes is found after a certain number of trials, which is essential for solving the given problem.

Probability Calculation

Probability calculation involves determining the likelihood of a specific event occurring, expressed as a number between 0 and 1. In this scenario, we need to calculate the probability that the first smoker is not among the first six selected adults, which requires understanding how to apply the geometric distribution formula effectively.

Unusual Events

An event is considered unusual if its probability is significantly low, often defined as less than 5%. In this exercise, after calculating the probability of the first smoker being outside the first six selected, we must assess whether this probability qualifies as unusual, providing insight into the behavior of smoking prevalence among the selected group.