Exercises 33 and 34 involve the method of composite sampling, whereby a medical testing laboratory saves time and money by combining blood samples for tests so that only one test is conducted for several people. A combined sample tests positive if at least one person has the disease. If a combined sample tests positive, then individual blood tests are used to identify the individual with the disease or disorder.


HIV It is estimated that in the United States, the proportion of people infected with the human immunodeficiency virus (HIV) is 0.00343. In tests for HIV, blood samples from 50 different people are combined. What is the probability that the combined sample tests positive for HIV? Is it unlikely for such a combined sample to test positive?

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to calculate the probability that a combined blood sample from 50 people tests positive for HIV. A combined sample tests positive if at least one person in the group is infected with HIV. The proportion of people infected with HIV in the United States is given as 0.00343.
2. Step 2: Define the probability of an individual not being infected with HIV. If the probability of being infected is 0.00343, then the probability of not being infected is calculated as 1 - 0.00343. This represents the complement of the infection probability.
3. Step 3: Calculate the probability that all 50 individuals in the combined sample are not infected. Since the infection status of each individual is independent, the probability that all 50 individuals are not infected is the product of the individual probabilities of not being infected. This can be expressed mathematically as \( P(\text{all not infected}) = (1 - 0.00343)^{50} \).
4. Step 4: Determine the probability that at least one person in the combined sample is infected. This is the complement of the probability that all 50 individuals are not infected. Mathematically, \( P(\text{at least one infected}) = 1 - P(\text{all not infected}) \).
5. Step 5: Interpret the result. Once the probability is calculated, compare it to a threshold (e.g., 0.05 for unlikely events) to determine whether it is unlikely for the combined sample to test positive. If the probability is very small, it suggests that it is unlikely for the combined sample to test positive.

Key Concepts

Probability

Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it helps determine the chance that at least one individual in a combined sample of 50 people is infected with HIV. Understanding how to calculate probabilities, especially in composite scenarios, is crucial for interpreting the results of the combined blood tests.

Binomial Distribution

The binomial distribution is a statistical distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this case, it can be used to calculate the probability of having at least one positive HIV test in a sample of 50 people, given the known infection rate. This concept is essential for understanding how to apply statistical methods to real-world testing scenarios.

Composite Sampling

Composite sampling is a technique where multiple samples are combined into one for testing, which can reduce costs and time in laboratory settings. This method is particularly useful in medical testing, as it allows for efficient screening of large populations. Understanding composite sampling is key to grasping how the combined test results can indicate the presence of a disease among individuals in the sample.