## **Core Concept**
The problem involves interpreting the results of a diagnostic test using its sensitivity, specificity, and the prevalence of the disease in the population. Sensitivity is the test's ability to correctly identify those with the disease (true positive rate), and specificity is its ability to correctly identify those without the disease (true negative rate).

## **Why the Correct Answer is Right**
To find the probability that a person negative to the test has no disease, we need to calculate the negative predictive value (NPV). The formula for NPV is:
[ NPV = frac{TN}{TN + FN} ]
Where:
- ( TN ) is the number of true negatives,
- ( FN ) is the number of false negatives.

First, let's establish the following:
- Prevalence of disease = 30% or 0.3,
- Sensitivity = 0.9,
- Specificity = 0.8.

Let's assume a population of 100 for simplicity:
- Diseased population = 30,
- Non-diseased population = 70.

For the diseased population (30):
- True positives (( TP )) = 0.9 * 30 = 27,
- False negatives (( FN )) = 30 - 27 = 3.

For the non-diseased population (70):
- True negatives (( TN )) = 0.8 * 70 = 56,
- False positives (( FP )) = 70 - 56 = 14.

The NPV is:
[ NPV = frac{56}{56 + 3} = frac{56}{59} ]

## **Why Each Wrong Option is Incorrect**
- **Option A:** This option is incorrect because it does not match our calculated NPV.
- **Option B:** This option is incorrect for the same reason; the calculation does not align with the provided numbers.
- **Option D:** This option is incorrect as it suggests a lower probability than calculated.

## **Clinical Pearl / High-Yield Fact**
A key point to remember is that the predictive values (positive and negative) of a test are influenced by the prevalence of the disease in the population being tested. In this case, even though the test has a good sensitivity and specificity, the NPV is not 100% due to the prevalence of the disease and the test's characteristics.

## **Correct Answer:** C. $frac{56}{59}$