A diagnostic test for a paicular disease has a sensitivity of 0.90 and a specificity of 0.90. A single test is applied to each subject in the population in which the diseased population is 10%. What is the probability that a person positive to this test, has the disease –

## Core Concept
The problem involves calculating the positive predictive value (PPV) of a diagnostic test. The PPV is the probability that a person with a positive test result actually has the disease. It depends on the test's sensitivity, specificity, and the disease prevalence in the population.

## Why the Correct Answer is Right
To calculate the PPV, we use the formula:
[ PPV = frac{Prevalence times Sensitivity}{(Prevalence times Sensitivity) + ((1 - Prevalence) times (1 - Specificity))} ]
Given:
- Sensitivity = 0.90
- Specificity = 0.90
- Prevalence = 10% or 0.10
Substituting these values:
[ PPV = frac{0.10 times 0.90}{(0.10 times 0.90) + ((1 - 0.10) times (1 - 0.90))} ]
[ PPV = frac{0.09}{(0.09) + (0.90 times 0.10)} ]
[ PPV = frac{0.09}{(0.09) + 0.09} ]
[ PPV = frac{0.09}{0.18} ]
[ PPV = 0.50 ]
So, the probability that a person positive to this test has the disease is 50%.

## Why Each Wrong Option is Incorrect
**Option A:** This option is incorrect because it does not match our calculated PPV of 0.50.
**Option B:** This option is incorrect for the same reason as Option A; the calculated PPV is 0.50, not a value that matches this option.
**Option D:** This option suggests a PPV of a different value, which does not align with our calculations.

## Clinical Pearl / High-Yield Fact
A key point to remember is that the positive predictive value (PPV) of a test is heavily influenced by the prevalence of the disease in the population being tested. In this case, even though the test has high sensitivity and specificity (both 0.90), the PPV is 0.50 because the disease prevalence is relatively low (10%). This highlights the importance of considering disease prevalence when interpreting test results.

## Correct Answer Line
**Correct Answer: C. 0.50**

✓ Correct Answer: B. 81%