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 (b) the fourth or fifth person selected.

검증된 단계별 안내

Step 1: Identify the appropriate probability distribution for the problem. Since we are looking for the probability that the first adult who smokes cigarettes is the fourth or fifth person selected, this is a geometric distribution problem. The geometric distribution models the probability of the first success occurring on a specific trial.

Step 2: Write the formula for the geometric distribution. The probability that the first success occurs on the nth trial is given by: P(X = n) = (1 - p)^(n-1) * p, where p is the probability of success (in this case, the probability that an adult smokes cigarettes), and (1 - p) is the probability of failure.

Step 3: Calculate the probability for the fourth person being the first smoker. Substitute n = 4 and p = 0.14 into the formula: P(X = 4) = (1 - 0.14)^(4-1) * 0.14. Simplify the expression to find the probability for the fourth person.

Step 4: Calculate the probability for the fifth person being the first smoker. Similarly, substitute n = 5 and p = 0.14 into the formula: P(X = 5) = (1 - 0.14)^(5-1) * 0.14. Simplify the expression to find the probability for the fifth person.

Step 5: Add the probabilities from Step 3 and Step 4 to find the total probability that the first smoker is either the fourth or fifth person. Finally, compare the result to a threshold (e.g., 0.05) to determine whether the event is unusual.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

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 applies to the scenario of finding the first adult who smokes cigarettes among a sample. The probability of success (smoking) is constant, and the trials continue until the first success occurs.

Probability Calculation

Calculating probabilities involves determining the likelihood of a specific event occurring. For the geometric distribution, the probability of the first success occurring on the k-th trial can be calculated using the formula P(X = k) = (1-p)^(k-1) * p, where p is the probability of success. This formula helps in finding the probability that the first smoker is the fourth or fifth person selected.

Unusual Events

An event is considered unusual if its probability is less than 0.05 (5%). In the context of this problem, after calculating the probabilities for the fourth and fifth adults being the first smokers, one must assess whether these probabilities fall below this threshold to determine if the events are unusual. This concept helps in interpreting the significance of the results.