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Chapter 5: Problem 15

False Positives The ELISA is a test to determine whether the HIV antibody is present. The test is \(99.5 \%\) effective, which means that the test will come back negative if the HIV antibody is not present \(99.5 \%\) of the time. The probability of a test coming back positive when the antibody is not present (a false positive) is \(0.005 .\) Suppose that the ELISA is given to five randomly selected people who do not have the HIV antibody. (a) What is the probability that the ELISA comes back negative for all five people? (b) What is the probability that the ELISA comes back positive for at least one of the five people?

Short Answer

Expert verified
The probability of all five tests being negative is approximately 97.52%. The probability of at least one test being positive is approximately 2.48%.

Step by step solution

01

- Identify the probability of a negative test

Since the test is 99.5% effective, the probability of getting a negative result if the person does not have the HIV antibody is 0.995.
02

- Determine the probability for all five tests

The tests are independent, so the probability that all five tests come back negative is the product of the individual probabilities:
03

- Calculate the probability for all five tests

Calculate
04

- Identify the probability of a positive test

The probability that at least one test comes back positive is the complement of all tests being negative. Use the formula:
05

- Calculate the complement

Calculate

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

False Positives
A 'false positive' occurs when a test indicates the presence of a condition, like HIV antibodies, even though the condition is not actually present. This can be concerning, especially for serious conditions, as it could lead to unnecessary stress and further testing. The probability of a false positive is a key metric in evaluating the effectiveness of any medical test.
Independent Events
In probability, events are considered 'independent' if the outcome of one event does not affect the outcome of another. For example, if you test five individuals for HIV using the ELISA test and none of them have HIV, each test result is independent of the others. This means the probability of each test coming back negative is not influenced by the results of the other tests. If the test is 99.5% effective, the probability of getting a negative result for each person is 0.995.
Complement Rule
The 'complement rule' is a fundamental concept in probability that helps calculate the likelihood of an event not occurring. If the probability of an event A occurring is denoted as P(A), then the probability of event A *not* occurring is given by 1 - P(A). For example, if the probability that all five ELISA tests come back negative is 0.995^5, the probability that at least one comes back positive is 1 - 0.995^5.
ELISA Test Effectiveness
The ELISA (Enzyme-Linked Immunosorbent Assay) test is designed to detect the presence of HIV antibodies with a high level of accuracy. An effectiveness rate of 99.5% means the test correctly identifies individuals without HIV antibodies 99.5% of the time. However, this also implies a 0.5% chance of a false positive, as discussed earlier. Understanding the test's effectiveness helps in interpreting the results and planning further medical evaluations.
HIV Testing in Statistics
HIV testing often involves statistical methods to understand the accuracy and reliability of different tests. Using probability, one can evaluate the likelihood of different outcomes, such as false positives or negatives. The ELISA test's effectiveness and the rare occurrence of false positives can be better understood through statistical analysis, which is crucial for public health measures and for individuals undergoing testing.

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Most popular questions from this chapter

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