Math Problem Statement
In July 2005, the journal Annals of Internal Medicine published a report on the reliability of HIV testing. Results of a large study suggested that among people with HIV, 96.4% of tests conducted were (correctly) positive, while for people without HIV 97.8% of the tests were (correctly) negative. A clinic serving an at-risk population offers HIV testing, believing that 26% of its patients may actually carry HIV. Note: You may use Tree Diagram to answer this question. (a) (4 points) What is the probability that a patient tested negative and is free of HIV?
Solution
To find the probability that a patient tested negative and is free of HIV, we use the Tree Diagram approach and calculate based on conditional probabilities. Here's the step-by-step solution:
Definitions
- Let : Probability that a patient has HIV.
- Let : Probability that a patient does not have HIV.
- Sensitivity = : Probability the test is positive given the patient has HIV.
- Specificity = : Probability the test is negative given the patient does not have HIV.
We need to find:
Step-by-Step Calculation
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Probability of no HIV:
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Probability of a negative test given no HIV:
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Probability of being negative and free of HIV: Substituting the values:
Final Answer
The probability that a patient tested negative and is free of HIV is:
Would you like a detailed explanation of how to build the tree diagram? Or do you want to continue solving related parts of this problem?
Related Questions:
- What is the probability a patient is tested positive but is free of HIV?
- What is the probability a patient is tested positive and has HIV?
- What is the probability a patient is tested negative and actually has HIV?
- What is the overall probability a patient tests negative?
- How does the sensitivity of a test affect the probability of false negatives?
Tip:
Always verify the sum of all branches of your probability tree equals 1 to ensure consistency in the calculations!
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Math Problem Analysis
Mathematical Concepts
Probability
Conditional Probability
Bayesian Analysis
Formulas
P(A and B) = P(A) * P(B|A)
P(Test Negative and No HIV) = P(No HIV) * P(Test Negative | No HIV)
Theorems
Law of Total Probability
Suitable Grade Level
Grades 11-12
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