Let's solve the problem using the given data and Bayes' Theorem.

Step 1: Organize the data

We are given that the total population is 100,000 people:

• 800 are infected (0.8% of the population)
• 99,200 are not infected (the remaining population)

We also have:

• The test has an 85% true positive rate (P(Positive Test | Infected) = 0.85)
• The test has a 12% false positive rate (P(Positive Test | Not Infected) = 0.12)

Now, let's fill out the contingency table.

Step 2: Calculate the number of people in each category

a) Positive Test:

• For infected individuals, 85% will test positive. So: $\text{Positive Test and Infected} = 800 \times 0.85 = 680$
• For not infected individuals, 12% will test positive. So: $\text{Positive Test and Not Infected} = 99,200 \times 0.12 = 11,904$

Total number of positive tests: $\text{Total Positive Tests} = 680 + 11,904 = 12,584$

b) Negative Test:

• For infected individuals, 15% will test negative (since 100% - 85% = 15%). So: $\text{Negative Test and Infected} = 800 \times 0.15 = 120$
• For not infected individuals, 88% will test negative (since 100% - 12% = 88%). So: $\text{Negative Test and Not Infected} = 99,200 \times 0.88 = 87,296$

Total number of negative tests: $\text{Total Negative Tests} = 120 + 87,296 = 87,416$

Step 3: Complete the table

	Infected	Not Infected	Total
Positive Test	680	11,904	12,584
Negative Test	120	87,296	87,416
Total	800	99,200	100,000

Step 4: Calculate the required probabilities

a) Probability that a person has the virus given that they have tested positive:

This is $P(\text{Infected} | \text{Positive Test})$, and we use the formula: $P(\text{Infected} | \text{Positive Test}) = \frac{P(\text{Infected and Positive Test})}{P(\text{Positive Test})}$ Substitute the values: $P(\text{Infected} | \text{Positive Test}) = \frac{680}{12,584} \approx 0.0541$

Rounding to the nearest tenth of a percent: $P(\text{Infected} | \text{Positive Test}) \approx 5.4\%$

b) Probability that a person does not have the virus given that they test negative:

This is $P(\text{Not Infected} | \text{Negative Test})$, and we use the formula: $P(\text{Not Infected} | \text{Negative Test}) = \frac{P(\text{Not Infected and Negative Test})}{P(\text{Negative Test})}$ Substitute the values: $P(\text{Not Infected} | \text{Negative Test}) = \frac{87,296}{87,416} \approx 0.9986$

Rounding to the nearest tenth of a percent: $P(\text{Not Infected} | \text{Negative Test}) \approx 99.9\%$

Final Answers:

• $P(\text{Infected} | \text{Positive Test}) = 5.4\%$
• $P(\text{Not Infected} | \text{Negative Test}) = 99.9\%$

Would you like further details or have any questions about the process?

Here are some related questions to explore:

1. How does changing the false positive rate affect the results?
2. How does the prevalence of the virus in the population affect the probabilities?
3. What would happen if the test had a 100% accuracy for infected individuals?
4. How would the results change if only 0.5% of the population were infected?
5. What are the implications of the false positive rate in large-scale testing?

Tip: The accuracy of a test depends not only on its sensitivity (true positive rate) but also on the prevalence of the condition in the population and the false positive rate.