Composite Drug Test Based on the data in Table 4-1, assume that the probability of a randomly selected person testing positive for drug use is 0.126. If drug screening samples are collected from 5 random subjects and combined, find the probability that the combined sample will reveal a positive result. Is that probability low enough so that further testing of the individual samples is rarely necessary?

Verified step by step guidance

1. Step 1: Understand the problem. The probability of a randomly selected person testing positive for drug use is given as 0.126. When samples from 5 individuals are combined, the combined sample will test positive if at least one of the individuals in the group tests positive. This is a complementary probability problem where we calculate the probability of the complement (all individuals testing negative) and subtract it from 1.
2. Step 2: Define the complement event. The complement event is that all 5 individuals test negative. The probability of a single individual testing negative is 1 - 0.126 = 0.874.
3. Step 3: Calculate the probability of all 5 individuals testing negative. Since the samples are independent, the probability of all 5 individuals testing negative is the product of the probabilities of each individual testing negative: (0.874)^5.
4. Step 4: Calculate the probability of the combined sample testing positive. The probability of the combined sample testing positive is the complement of all individuals testing negative, which is 1 - (0.874)^5.
5. Step 5: Interpret the result. Once the probability is calculated, compare it to a threshold (e.g., 0.05) to determine if it is low enough to justify further testing of individual samples. If the probability is low, further testing may not be necessary. If it is not low, further testing may be required.

Key Concepts

Probability of Events

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, the probability of a randomly selected person testing positive for drug use is given as 0.126, indicating a 12.6% chance of a positive result. Understanding how to calculate the probability of combined events is crucial for determining the likelihood of a positive result from multiple samples.

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 scenario, we can use the binomial distribution to find the probability of at least one positive test result when combining samples from 5 subjects, where each subject has a 12.6% chance of testing positive. This concept is essential for calculating the overall probability of a positive outcome.

Complement Rule

The complement rule states that the probability of an event occurring is equal to 1 minus the probability of it not occurring. In this case, to find the probability that at least one of the combined samples tests positive, we can first calculate the probability that none of the samples test positive and then subtract that from 1. This approach simplifies the calculation and helps assess whether further testing is necessary.