To solve for $P(\text{Has Virus | Tested Positive})$, we can use Bayes' Theorem, which states:

$P(\text{A | B}) = \frac{P(\text{B | A}) \cdot P(\text{A})}{P(\text{B})}$

where:

• $P(\text{A})$ is the probability that a person has the virus.
• $P(\text{B | A})$ is the probability that the test is positive given that the person has the virus.
• $P(\text{B})$ is the probability that the test is positive.

Given values:

1. The probability that a person has the virus, $P(\text{A}) = \frac{1}{300}$.
2. The probability that a test is positive given the person has the virus, $P(\text{B | A}) = 0.8$.
3. The probability of a false positive, $P(\text{B | Not A}) = 0.05$.

We first need to calculate $P(\text{B})$, the probability that a person tests positive. This can happen in two ways:

1. The person has the virus and tests positive.
2. The person does not have the virus but still tests positive (false positive).

Thus,

$P(\text{B}) = P(\text{B | A}) \cdot P(\text{A}) + P(\text{B | Not A}) \cdot P(\text{Not A})$

where:

• $P(\text{Not A}) = 1 - P(\text{A})$.

Step-by-Step Calculation

1. Calculate $P(\text{A})$:

   $P(\text{A}) = \frac{1}{300} \approx 0.00333$

2. Calculate $P(\text{Not A})$:

   $P(\text{Not A}) = 1 - 0.00333 = 0.99667$

3. Calculate $P(\text{B})$:

   $P(\text{B}) = (0.8 \times 0.00333) + (0.05 \times 0.99667)$ $P(\text{B}) = 0.002664 + 0.0498335 = 0.0524975$

4. Apply Bayes' Theorem to find $P(\text{A | B})$:

   $P(\text{A | B}) = \frac{P(\text{B | A}) \cdot P(\text{A})}{P(\text{B})}$ $P(\text{A | B}) = \frac{0.8 \times 0.00333}{0.0524975}$ $P(\text{A | B}) \approx \frac{0.002664}{0.0524975} \approx 0.0508$

5. Convert to a percentage and round to the nearest tenth of a percent:

   $P(\text{A | B}) \approx 5.1\%$

Final Answer:

$P(\text{Has Virus | Tested Positive}) = 5.1\%$

Would you like further details on any steps, or do you have questions?

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Here are five related questions:

1. What is the probability of a false positive in this test scenario?
2. How does a higher accuracy in the test affect the conditional probability?
3. If the infection rate increases to 1 in 100, how does $P(\text{Has Virus | Tested Positive})$ change?
4. How can false negatives impact the test results?
5. What is Bayes' Theorem, and where else is it used?

Tip: When interpreting conditional probabilities, remember that a low prevalence rate can significantly affect the predictive value of a positive test result.