Problem 9
Question
Suppose the blood of 1000 persons has to be tested to see which ones are infected by a (rare) disease. Suppose that the probability that the test is positive is \(p=0.001\). The obvious way to proceed is to test each person, which results in a total of 1000 tests. An alternative procedure is the following. Distribute the blood of the 1000 persons over 25 groups of size 40 , and mix half of the blood of each of the 40 persons with that of the others in each group. Now test the aggregated blood sample of each group: when the test is negative no one in that group has the disease; when the test is positive, at least one person in the group has the disease, and one will test the other half of the blood of all 40 persons of that group separately. In total, that gives 41 tests for that group. Let \(X_{i}\) be the total number of tests one has to perform for the \(i\) th group using this alternative procedure. a. Describe the probability distribution of \(X_{i}\), i.e., list the possible values it takes on and the corresponding probabilities. b. What is the expected number of tests for the \(i\) th group? What is the expected total number of tests? What do you think of this alternative procedure for blood testing?
Step-by-Step Solution
Verified Answer
A. Possible values: 1 (probability \((0.999)^{40}\)) and 41; B. Expected group tests: approximately 3; expected total tests: around 75.
1Step 1: Understanding the Problem
We have 1000 individuals that need to be tested for a disease, with a probability of a positive individual being 0.001. The objective is to assess an alternative testing method by grouping individuals and testing them in batches.
2Step 2: Defining Possible Outcomes for Xi
For a group of 40 people, there are two possible outcomes:
1. The group test is negative: no individual tests are needed, resulting in a total of 1 test.
2. The group test is positive: all individuals need to be tested, resulting in 40 additional tests, for a total of 41 tests.
3Step 3: Probability of Negative Group Test
We calculate the probability that the aggregated test result for a group is negative. The probability that one person does not have the disease is 0.999. Hence, the probability that none of the 40 individuals in a group test positive is \( (0.999)^{40} \).
4Step 4: Probability of Positive Group Test
The probability that at least one person in the group has the disease (and thus the group test is positive) is the complement of the previous step: \( 1 - (0.999)^{40} \).
5Step 5: Calculating Expected Value of Xi
The expected number of tests \( E(X_i) \) for a group is calculated using the formula:\[ E(X_i) = 1 \times P(\text{negative group test}) + 41 \times P(\text{positive group test}) \]Substitute the probabilities from previous steps:\[ E(X_i) = 1 \times (0.999)^{40} + 41 \times (1 - (0.999)^{40}) \]
6Step 6: Expected Total Number of Tests
The expected total number of tests for all 25 groups is 25 times the expected number of tests for one group: \[ E(\text{total tests}) = 25 \times E(X_i) \].Calculate using the expected value from Step 5.
7Step 7: Evaluating the Alternative Procedure
Compare the expected total number of tests using the alternative method to the original method (1000 tests). The decision on the effectiveness of the procedure depends on whether the expected tests in the alternative method are fewer than 1000.
Key Concepts
Alternative Testing MethodExpected ValueProbability CalculationsGroup Testing Strategy
Alternative Testing Method
An alternative testing method can offer substantial benefits in terms of efficiency and cost savings when dealing with large groups of samples, such as blood tests for a rare disease. In this context, rather than testing each individual's blood sample separately, which would require 1000 tests for 1000 people, a more strategic approach involves grouping samples and initially testing them in batches.
By aggregating the blood of 40 individuals into one test, we drastically reduce the number of tests needed. This method works particularly well for rare diseases, where the chance of finding a positive case in any given group is quite low. If the group test is negative, we can confidently say no individuals within that group are infected. However, if the group test is positive, further individual tests are required to pinpoint the infected person or persons. This innovation exemplifies thinking outside the box in medical testing, providing a feasible and potentially more resource-efficient solution.
By aggregating the blood of 40 individuals into one test, we drastically reduce the number of tests needed. This method works particularly well for rare diseases, where the chance of finding a positive case in any given group is quite low. If the group test is negative, we can confidently say no individuals within that group are infected. However, if the group test is positive, further individual tests are required to pinpoint the infected person or persons. This innovation exemplifies thinking outside the box in medical testing, providing a feasible and potentially more resource-efficient solution.
Expected Value
The expected value is a foundational concept in probability, representing the average outcome if an experiment is repeated multiple times. In the case of our alternative testing method, calculating the expected number of tests can help us understand the efficiency of this method.
For a single group of 40 individuals, there are two potential outcomes: a negative group test or a positive group test. The expected value for the number of tests, denoted as \( E(X_i) \), combines both possible outcomes weighted by their probabilities:
This approach clarifies how often, on average, additional individual tests are needed following a group test.
For a single group of 40 individuals, there are two potential outcomes: a negative group test or a positive group test. The expected value for the number of tests, denoted as \( E(X_i) \), combines both possible outcomes weighted by their probabilities:
- A negative test for the group results in just 1 test, occurring with a probability of \( (0.999)^{40} \).
- A positive test results in 41 tests (1 group test plus 40 individual tests), with a probability of \( 1 - (0.999)^{40} \).
This approach clarifies how often, on average, additional individual tests are needed following a group test.
Probability Calculations
Probability calculations play a critical role in determining the effectiveness of a group testing strategy. For a group of 40 people, understanding both the probability of a negative and positive group test is pivotal.
To begin, the probability that a single person does not have the disease is 0.999. Since the disease is rare, the likelihood of none of the 40 people in a group having the disease is calculated using multiplication for independent events: \( (0.999)^{40} \). This is the probability of a negative group test.
Conversely, the probability of at least one person having the disease is the complement of the above, calculated as \( 1 - (0.999)^{40} \). This value tells us how often a follow-up with 41 tests will be necessary due to a positive outcome for the group test. By mastering these simple probability calculations, complex questions about testing strategies become manageable and enlightening.
To begin, the probability that a single person does not have the disease is 0.999. Since the disease is rare, the likelihood of none of the 40 people in a group having the disease is calculated using multiplication for independent events: \( (0.999)^{40} \). This is the probability of a negative group test.
Conversely, the probability of at least one person having the disease is the complement of the above, calculated as \( 1 - (0.999)^{40} \). This value tells us how often a follow-up with 41 tests will be necessary due to a positive outcome for the group test. By mastering these simple probability calculations, complex questions about testing strategies become manageable and enlightening.
Group Testing Strategy
A group testing strategy can significantly optimize resources, particularly when dealing with a disease of low prevalence. By testing in groups, we harness the power of probability and expected outcomes to reduce total tests required.
Initially, dividing 1000 blood samples into 25 groups of 40 each allows us to conduct fewer initial tests. For each group, if the batch test is negative, only 1 test is needed. However, a positive test necessitates individual testing, leading to a total of 41 tests for that group.
The benefits of this strategy are apparent when calculating the expected total number of tests across all groups. By assessing the expected value for one group's tests (as previously discussed) and then multiplying by the number of groups, we achieve an anticipated total number of tests needed. This strategy, therefore, balances the likelihood of positive and negative results to minimize the overall testing burden. This approach not only saves time but also resources, making it a favored choice in large-scale testing scenarios.
Initially, dividing 1000 blood samples into 25 groups of 40 each allows us to conduct fewer initial tests. For each group, if the batch test is negative, only 1 test is needed. However, a positive test necessitates individual testing, leading to a total of 41 tests for that group.
The benefits of this strategy are apparent when calculating the expected total number of tests across all groups. By assessing the expected value for one group's tests (as previously discussed) and then multiplying by the number of groups, we achieve an anticipated total number of tests needed. This strategy, therefore, balances the likelihood of positive and negative results to minimize the overall testing burden. This approach not only saves time but also resources, making it a favored choice in large-scale testing scenarios.
Other exercises in this chapter
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