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.

검증된 단계별 안내

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

Step 2: Identify the parameters of the geometric distribution. The probability of success (smoking) is given as \( p = 0.14 \), and the probability of failure (not smoking) is \( q = 1 - p = 0.86 \).

Step 3: Write the formula for the geometric distribution. The probability that the first success occurs on the \( k \)-th trial is given by \( P(X = k) = q^{k-1} \cdot p \), where \( k \) is the trial number.

Step 4: Substitute the given values into the formula. For \( k = 3 \), the probability is \( P(X = 3) = q^{3-1} \cdot p = q^2 \cdot p = (0.86)^2 \cdot 0.14 \).

Step 5: Determine whether the event is unusual. An event is typically considered unusual if its probability is less than 0.05. Compare the calculated probability to 0.05 to make this determination.

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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. In this context, it is used to find the probability that the first adult who smokes is the third person selected, where each trial (asking an adult) has a constant probability of success (14% smoking rate).

Probability Calculation

Calculating probabilities involves determining the likelihood of a specific outcome occurring. For the geometric distribution, the probability of the first success occurring on the k-th trial is given by the formula P(X = k) = (1-p)^(k-1) * p, where p is the probability of success. This formula is essential for solving the given problem.

Unusual Events

An event is considered unusual if its probability is low, typically defined as less than 5%. In this exercise, after calculating the probability of the first smoker being the third person selected, one must assess whether this probability qualifies as unusual, providing insight into the likelihood of such an occurrence in the context of the population.