A certain virus infects one in every 500 people. A test used to detect the virus in a person is positive $90 \%$ of the time if the person has the virus and $10 \%$ of the time if the person does not have the virus. Let $A$ be the event "the person is infected" and B be the event "the person tests positive." (a) Find the probability that a person has the virus given that they have tested positive. (b) Find the probability that a person does not have the virus given that they have tested negative.

Step 1 ：Let's denote the event 'the person is infected' as A and the event 'the person tests positive' as B.

Step 2 ：We are asked to find two conditional probabilities: P(A|B), the probability that a person has the virus given that they have tested positive, and P(A'|B'), the probability that a person does not have the virus given that they have tested negative.

Step 3 ：We can use Bayes' theorem to find these probabilities. Bayes' theorem states that P(A|B) = P(B|A)P(A) / P(B) and P(A'|B') = P(B'|A')P(A') / P(B').

Step 4 ：We know that P(A), the probability that a person has the virus, is 1/500. Therefore, P(A'), the probability that a person does not have the virus, is 1 - 1/500 = 499/500.

Step 5 ：We also know that P(B|A), the probability that a person tests positive given that they have the virus, is 90%. Therefore, P(B'|A'), the probability that a person tests negative given that they do not have the virus, is 1 - 10% = 90%.

Step 6 ：We can find P(B), the probability that a person tests positive, and P(B'), the probability that a person tests negative, using the law of total probability: P(B) = P(B|A)P(A) + P(B|A')P(A') and P(B') = P(B'|A)P(A) + P(B'|A')P(A').

Step 7 ：Substituting the known values into Bayes' theorem, we find that P(A|B) = 0.0177 and P(A'|B') = 0.9998.

Step 8 ：Final Answer: (a) The probability that a person has the virus given that they have tested positive is approximately \(\boxed{0.0177}\) or \(\boxed{1.77\%}\). (b) The probability that a person does not have the virus given that they have tested negative is approximately \(\boxed{0.9998}\) or \(\boxed{99.98\%}\).