First, let's analyze the given table to understand the data provided:
[tex]\[
\begin{array}{lccc}
& \text{Positive Test} & \text{Negative Test} & \text{Total} \\
\text{Has Disease} & 18 & 2 & 20 \\
\text{Does Not Have Disease} & 98 & 882 & 980 \\
\text{Total} & 116 & 884 & 1000 \\
\end{array}
\][/tex]
We need to determine the probability that a person does not have the disease given that they received a positive test result.
### Steps to Solve:
1. Identify the relevant numbers:
- Number of positive test results = 116 (from the total positive tests column).
- Number of positive test results where the person does not have the disease = 98 (from the "Does Not Have Disease" row under the positive test column).
2. Calculate the probability:
- The probability of not having the disease given a positive test result is the ratio of the number of positive tests where the person does not have the disease to the total number of positive tests.
[tex]\[
P(\text{No Disease} | \text{Positive Test}) = \frac{\text{Number of Positive Tests with No Disease}}{\text{Total Number of Positive Tests}} = \frac{98}{116}
\][/tex]
3. Convert the probability to a percentage:
- To convert this probability to a percentage, multiply by 100.
[tex]\[
P(\text{No Disease} | \text{Positive Test}) \times 100 = \left(\frac{98}{116}\right) \times 100
\][/tex]
4. Determine the final value:
- Given our calculations:
[tex]\[
\frac{98}{116} \approx 0.8448275862068966
\][/tex]
- Converting this to a percentage:
[tex]\[
0.8448275862068966 \times 100 \approx 84.48275862068965 \%
\][/tex]
### Conclusion:
The probability that a person does not have the disease given that they received a positive test result is approximately [tex]\( 84.5\% \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{84.5\%} \][/tex]
