Question

A laboratory blood test is 99% effective in detecting a certain disease when it is in fact present. However, the test also yields a false positive result for 0.5% of the healthy person tested. If 0.1% of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

Hint:

Even if a test is 99% accurate, if the disease prevalence is very low (like 0.1%), the probability of having the disease given a positive test can be surprisingly small.

Answer 1

Step 1: Understanding the Concept:
This problem is solved using Bayes' Theorem. We define events for having the disease and having a positive test result.
Step 2: Key Formula or Approach:
\[ P(D|+) = \frac{P(+|D)P(D)}{P(+|D)P(D) + P(+|H)P(H)} \]
Step 3: Detailed Explanation:
Let \( D \) be the event that the person has the disease.
Let \( H \) be the event that the person is healthy (no disease).
Let \( + \) be the event that the test result is positive.
Given:
\( P(D) = 0.1% = 0.001 \)
\( P(H) = 1 - 0.001 = 0.999 \)
\( P(+|D) = 99% = 0.99 \) (True positive)
\( P(+|H) = 0.5% = 0.005 \) (False positive)
Using Bayes' Theorem:
\[ P(D|+) = \frac{0.99 \times 0.001}{(0.99 \times 0.001) + (0.005 \times 0.999)} \]
\[ P(D|+) = \frac{0.00099}{0.00099 + 0.004995} \]
\[ P(D|+) = \frac{0.00099}{0.005985} \approx 0.1654 \]
Step 4: Final Answer:
The probability is approximately \( 0.165 \).