Explore a diverse range of topics and get expert answers on IDNLearn.com. Join our community to receive prompt and reliable responses to your questions from knowledgeable professionals.

The table below shows the predicted testing outcome for a disease that affects [tex]2\%[/tex] of a population and has a [tex]90\%[/tex] accuracy rate.

\begin{tabular}{lccc}
& Positive Test & Negative Test & Total \\
Has Disease & 18 & 2 & 20 \\
Does Not Have Disease & 98 & 882 & 980 \\
Total & 116 & 884 & 1000
\end{tabular}

Given you received a positive test, which of the following represents the probability that you do not have the disease?

A. [tex]84.5\%[/tex]

B. [tex]99.8\%[/tex]

C. [tex]15.5\%[/tex]

D. [tex]0.2\%[/tex]


Sagot :

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]