Problem 42 - Calculus for Biology and Medicine

Question

To test for a disease that has a prevalence of 1 in 100 in a population, blood samples of 10 individuals are pooled and the pooled blood is then tested. What is the probability that the test result is negative (the disease is not present in the pooled blood sample)?

Step-by-Step Solution

Answer

The probability that the pooled test result is negative is approximately 0.9044.

Step 1: Understanding the Prevalence

To determine the probability that the pooled blood sample does not contain the disease, we first need to understand the given prevalence. The disease has a prevalence of 1 in 100, meaning that the probability of one individual having the disease is \(P(H) = \frac{1}{100} = 0.01\). This means that the probability of one individual not having the disease is \(P(NH) = 1 - 0.01 = 0.99\).

Step 2: Probability of All Individuals Being Healthy

Since a pooled sample is negative only if all individuals in that pool are healthy, we need to calculate the probability that all 10 individuals do not have the disease. Each individual's probability of not having the disease is 0.99, and we assume independence between individuals. The probability that all 10 are healthy is given by raising the probability of one individual being healthy to the power of 10:\[P(All \, Healthy) = P(NH)^{10} = 0.99^{10}.\]

Step 3: Calculate the Probability

Perform the calculation for \(0.99^{10}\): \[0.99^{10} \approx 0.9044.\] This is the probability that none of the 10 individuals in the pooled sample have the disease, meaning the test would be negative.

Key Concepts

Understanding Prevalence

Prevalence refers to the proportion of a population found to have a specific condition at a given time. In probability theory, when discussing diseases, prevalence indicates the likelihood or probability that an individual randomly selected from the population has the disease. In our specific exercise, the prevalence of the disease is stated as 1 in 100. This means the probability of an individual having the disease, denoted as \(P(H)\), is \(\frac{1}{100} = 0.01\). Therefore, if you select a person randomly, there's a 1% chance this person has the disease. Correspondingly, the probability that an individual does not have the disease is \(P(NH) = 1 - 0.01 = 0.99\), or 99%. In simple terms, for every 100 individuals, we expect 99 to be disease-free. Prevalence helps us understand the base likelihood of encountering the disease within a given population.

Exploring the Independence Assumption

When working with probability, the independence assumption is crucial. Essentially, it assumes that the presence or absence of disease in one individual does not affect the others in the pool. Probability theory allows us to calculate combined probabilities for independent events by multiplying their individual probabilities. In the context of our exercise, the independence assumption means that knowing the health status of one individual in the pooled sample doesn’t change the likelihood of the others' statuses. Thus, for our 10 pooled individuals, the probability that each one is healthy is \(P(NH) = 0.99\). The combined probability that all 10 individuals are healthy is the product of each individual probability: \(0.99^{10}\), which calculates to approximately 0.9044. This assumption simplifies our calculations but also highlights a key limitation: if individuals' disease statuses are somehow correlated, the calculation would be different.

Mechanics of Pooled Testing

Pooled testing is an efficient method for testing a large number of samples for rare conditions. Instead of testing each individual sample separately, samples are combined, or pooled, into groups. The pooled sample is tested as a single entity. If the test is negative, we conclude that all individuals in the pool are disease-free. If positive, further testing is required to identify the affected individuals. The main advantage of pooled testing is cost and resource efficiency, particularly in populations where low prevalence is expected. In our exercise about a disease with a prevalence of \(1\) in \(100\), pooled testing is particularly beneficial. Here, we pooled 10 samples together, and through our calculations, found that there’s approximately a \(90.44\)% probability that the pooled test will be negative, meaning all individuals are healthy. This approach significantly reduces the number of tests needed, which is crucial for resource-saving, especially during mass screening scenarios.