Question

Kit Performance in an Outbreak
A researcher compiled the following information about the performance of a kit in an outbreak:

Infection State	Kit Response
Disease (probability = 0.002)	Positive response (probability = 0.98)
No Disease	Positive response (probability = 0.03)

The probability of detecting an infection for a positive result through the kit would be ______________________ (rounded off to three decimal places).

Hint:

In medical testing, even highly accurate kits may have very low predictive values when the disease prevalence is extremely low. Always apply Bayes’ theorem to interpret diagnostic probabilities.

Answer 1

Correct Answer: 0.06

Step 1: Recall Bayes’ theorem.
The probability of having the disease given that the kit is positive is: \[ P(D|+) = \frac{P(+|D) . P(D)}{P(+|D) . P(D) + P(+|\bar{D}) . P(\bar{D})} \] where - $P(D)$ = probability of having the disease, - $P(\bar{D})$ = probability of not having the disease, - $P(+|D)$ = probability of a positive response given disease, - $P(+|\bar{D})$ = probability of a positive response without disease.
Step 2: Substitute the values.
\[ P(D) = 0.002, P(\bar{D}) = 1 - 0.002 = 0.998 \] \[ P(+|D) = 0.98, P(+|\bar{D}) = 0.03 \]
Step 3: Numerator (true positive contribution).
\[ P(+|D) . P(D) = 0.98 \times 0.002 = 0.00196 \]
Step 4: Denominator (total probability of positive).
\[ = (0.98 \times 0.002) + (0.03 \times 0.998) = 0.00196 + 0.02994 = 0.0319 \]
Step 5: Calculate.
\[ P(D|+) = \frac{0.00196}{0.0319} \approx 0.0614 \] Rounded to three decimal places: $0.061$. Final Answer: \[ \boxed{0.061} \]