To solve this problem, we need to determine the conditional probability that the test result is positive given that the sample does not contain the bacteria.
We'll use the following steps:
1. Identify the relevant events:
- Event A: The sample does not contain bacteria.
- Event B: The test is positive.
2. Extract given data from the table:
- The number of samples that do not contain bacteria and test positive (Event A and Event B both occur): 58
- The total number of samples that do not contain bacteria (Event A): 1,930
3. Set up the formula for conditional probability:
[tex]\[
P(B|A) = \frac{P(A \cap B)}{P(A)}
\][/tex]
Here, [tex]\(P(A \cap B)\)[/tex] represents the number of samples with Event A and Event B both occurring, and [tex]\(P(A)\)[/tex] represents the number of samples with Event A.
4. Substitute the values into the formula:
[tex]\[
P(B|A) = \frac{58}{1,930}
\][/tex]
5. Calculate the probability:
[tex]\[
P(B|A) \approx 0.03005181347150259
\][/tex]
After performing the calculations, the probability that the test result is positive given that the sample does not contain the bacteria is approximately [tex]\(0.0301\)[/tex], which corresponds to approximately 3.01%.
Given the answer choices:
A. 0.001
B. 0.46
C. 0.00
D. 0.54
None of the given choices exactly match the calculated probability. Given this, it seems there may be an error in the provided answer choices. However, if we consider significant figures or a potential typo, the closest answer could be:
A. 0.001
But, strictly speaking, none of the provided answers exactly match the calculated probability of approximately 0.0301 (3.01%).
