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 P(HIV)=0.26P(\text{HIV}) = 0.26: Probability that a patient has HIV.
  • Let P(No HIV)=0.74P(\text{No HIV}) = 0.74: Probability that a patient does not have HIV.
  • Sensitivity = P(Test Positive | HIV)=0.964P(\text{Test Positive | HIV}) = 0.964: Probability the test is positive given the patient has HIV.
  • Specificity = P(Test Negative | No HIV)=0.978P(\text{Test Negative | No HIV}) = 0.978: Probability the test is negative given the patient does not have HIV.

We need to find: P(Negative and No HIV)=P(No HIV)P(Test Negative | No HIV)P(\text{Negative and No HIV}) = P(\text{No HIV}) \cdot P(\text{Test Negative | No HIV})


Step-by-Step Calculation

  1. Probability of no HIV: P(No HIV)=0.74P(\text{No HIV}) = 0.74

  2. Probability of a negative test given no HIV: P(Test Negative | No HIV)=0.978P(\text{Test Negative | No HIV}) = 0.978

  3. Probability of being negative and free of HIV: P(Negative and No HIV)=P(No HIV)P(Test Negative | No HIV)P(\text{Negative and No HIV}) = P(\text{No HIV}) \cdot P(\text{Test Negative | No HIV}) Substituting the values: P(Negative and No HIV)=0.740.978=0.72372P(\text{Negative and No HIV}) = 0.74 \cdot 0.978 = 0.72372


Final Answer

The probability that a patient tested negative and is free of HIV is: P(Negative and No HIV)=0.72372or approximately 72.4%.P(\text{Negative and No HIV}) = 0.72372 \, \text{or approximately } 72.4\%.


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:

  1. What is the probability a patient is tested positive but is free of HIV?
  2. What is the probability a patient is tested positive and has HIV?
  3. What is the probability a patient is tested negative and actually has HIV?
  4. What is the overall probability a patient tests negative?
  5. 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