## **Core Concept**
The problem involves calculating the sensitivity of a diagnostic test. Sensitivity, also known as the true positive rate, is the proportion of actual positives that are correctly identified by the test. It is calculated as the number of true positive results divided by the sum of true positives and false negatives.

## **Why the Correct Answer is Right**
To calculate the sensitivity, we first need to understand the given data:
- True positives (TP) + False positives (FP) = 80 (total positive cases)
- False positives (FP) = 40 (cases with no disease but tested positive)
- True negatives (TN) + False negatives (FN) = 9920 (total negative cases)
- True negatives (TN) = 9840 (cases with no disease and tested negative)

From the given data:
- True positives (TP) = 80 - 40 = 40
- False negatives (FN) = 9920 - 9840 = 80

The sensitivity of the test is calculated as:
[ text{Sensitivity} = frac{TP}{TP + FN} = frac{40}{40 + 80} = frac{40}{120} = frac{1}{3} ]

## **Why Each Wrong Option is Incorrect**
- **Option A:** This option does not match our calculated sensitivity.
- **Option B:** This option suggests a sensitivity of 1/2 or 50%, which does not align with our calculation of 1/3.
- **Option D:** This option suggests a much higher sensitivity than our calculated value.

## **Clinical Pearl / High-Yield Fact**
A key point to remember is that sensitivity and specificity are crucial metrics for evaluating the performance of a diagnostic test. Sensitivity indicates the test's ability to detect those with the disease, while specificity indicates its ability to exclude those without the disease. A highly sensitive test is useful for ruling out a disease when the test is negative.

## **Correct Answer: C. 1/3**