Question

To reduce the cost of detecting a disease, blood tests are conducted on a pooled sample of blood collected from a group of \(n\) people. If no indication of the disease is present in the pooled blood sample, none have the disease. If analysis of the pooled blood sample indicates that the disease is present, the blood of each individual must be tested. The individual tests are conducted in sequence. If, among a group of five people, one person has the disease, what is the probability that six blood tests (including the pooled test) are required to detect the single diseased person? If two people have the disease, what is the probability that six tests are required to locate both diseased people?

Short answer

Answer: The probability of needing 6 blood tests to detect one diseased person among five is 1/5. The probability of needing 6 tests to locate both diseased people when there are 2 diseased individuals among the five is also 1/5.

Step-by-step solution

Part A: Probability with one diseased individual

In this case, there is only one diseased person among the group of five. We need six blood tests, including the pooled test, to detect the disease. This means that the diseased person must be the last one to get tested individually. There are 5 people in the group, so the probability of the diseased person being the last one to get tested is \(\frac{1}{5}\).

Part B: Probability with two diseased individuals

Now, there are two diseased individuals among the group of five. We need six blood tests, including the pooled test, to locate both diseased people. This means that the two diseased individuals must be the last two to get tested individually. We need to find the number of ways to arrange the testing sequence such that the last two are the diseased individuals. There are \(\binom{5}{2} = 10\) possible ways to choose which two people are the diseased individuals. In each of these 10 cases, the last two tested must be the diseased individuals. So, there are a total of 3 permutations for placing the two diseased individuals at the end: \((D_1, D_2)\), \((D_2, D_1)\), and \((D_3, D_4)\). Out of these three permutations, only \((D_1, D_2)\) and \((D_2, D_1)\) are valid since they have both diseased individuals. The probability of either of these two cases happening is \(\frac{1}{\binom{5}{2}} = \frac{1}{10}\) for each case. So, the required probability is the sum of the probabilities of these two cases, which is \(\frac{1}{10} + \frac{1}{10} = \frac{2}{10} = \frac{1}{5}\).

Key concepts

Pooled Sampling

Pooling samples in medical testing is a method used to reduce costs and resources. Instead of testing each blood sample individually, blood from multiple people is combined in one test. This economizes laboratory resources and speeds up the process when testing for a disease. However, the efficiency of pooled sampling relies heavily on the prevalence of the disease within the population. If no disease is detected in the pooled sample, there is no need for further tests on individuals, implying all are disease-free. But if the pooled sample indicates disease presence, each individual's sample must be tested separately to identify the infected persons. This can sometimes lead to extensive testing if not managed effectively.

Disease Detection

The primary goal of disease detection is to accurately identify the presence of disease in individuals. When pooled sampling is used and a pooled sample tests positive, it's confirmed that at least one individual carries the disease. Consequently, individual tests become essential to determine who is affected. Efficient disease detection in this scenario focuses on minimizing the number of tests while ensuring reliability. Accurately determining infected individuals requires a delicate balance between cost-efficiency and thoroughness, especially in groups with varied disease prevalence.

Probability Calculation

Probability is a key element in understanding outcomes in pooled sampling. When dealing with groups of people, the probability can be used to predict events like the number of tests needed based on group size and number of diseased individuals. For example, if there's one diseased person among a group of five, the probability that six tests are conducted (including pooled and individual) is determined by the likelihood that this person is the last one individually tested. This probability is calculated by focusing on the order or sequence of testing, which becomes more intricate if more individuals in the group have the disease. Thus, understanding the fundamental principles of probability helps in forecasting testing requirements efficiently.

Permutation and Combination

Permutation and combination are mathematical methods used to calculate the different ways or orders in which members of a set can be arranged. In the context of disease detection using pooled sampling, permutations help determine the order in which tests are done if multiple people might have a disease. For example, if there are two diseased individuals in a sample of five, permutations calculate how the sequence of tests can end with these two individuals. Combinations, on the other hand, help determine how diseased individuals are selected out of the group. Understanding these concepts allows researchers to strategically plan testing procedures and accurately estimate probabilities in disease detection scenarios.