In Exercises 1–5, assume that 4.2% of workers test positive when tested for illegal drugs (based on data from Quest Diagnostics). Assume that a group of ten workers is randomly selected.

Workplace Drug Testing Find the probability that exactly two of the ten workers test positive for illegal drugs.

검증된 단계별 안내

Step 1: Recognize that this is a binomial probability problem. The binomial distribution is used because there are a fixed number of trials (n = 10 workers), each trial has two possible outcomes (positive or negative drug test), the probability of success (testing positive) is constant (p = 0.042), and the trials are independent.

Step 2: Write the formula for the binomial probability: P(X = k) = (n choose k) * p^k * (1 - p)^(n - k), where 'n' is the number of trials, 'k' is the number of successes, 'p' is the probability of success, and '1 - p' is the probability of failure.

Step 3: Substitute the given values into the formula. Here, n = 10, k = 2, and p = 0.042. The formula becomes: P(X = 2) = (10 choose 2) * (0.042)^2 * (1 - 0.042)^(10 - 2).

Step 4: Calculate the binomial coefficient (10 choose 2), which is given by the formula: (n choose k) = n! / [k! * (n - k)!]. For this problem, (10 choose 2) = 10! / [2! * (10 - 2)!].

Step 5: Simplify the remaining terms: (0.042)^2 represents the probability of two workers testing positive, and (1 - 0.042)^8 represents the probability of the remaining eight workers testing negative. Multiply all these components together to find the probability.

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.

Binomial Probability

Binomial probability refers to the probability of obtaining a fixed number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, testing positive for illegal drugs is considered a 'success,' and the scenario involves ten independent tests, making it suitable for binomial distribution analysis.

Probability of Success and Failure

In a binomial distribution, the probability of success (testing positive) is denoted as 'p,' while the probability of failure (testing negative) is 'q,' where q = 1 - p. For this problem, p is 0.042 (4.2%), and q is 0.958 (95.8%). Understanding these probabilities is crucial for calculating the likelihood of different outcomes.

Combination Formula

The combination formula, denoted as C(n, k) or nCk, calculates the number of ways to choose k successes from n trials without regard to the order of selection. This is essential for determining how many different groups of two workers can test positive out of ten, which is a key component in calculating the overall probability of exactly two positives.