Question: The New York State Health Department reports a 10% rate of the HIV virus for the “at-risk” population. Under certain conditions, a preliminary screening test for the HIV virus is correct 95% of the time. (Subjects are not told that they are HIV infected until additional tests verify the results.) If someone is randomly selected from the at-risk population, what is the probability that they have the HIV virus if it is known that they have tested positive in the initial screening? 10. HIV Use the same data from Exercise 9. If someone is randomly selected from the at-risk population, what is the probability that they have the HIV virus if it is known that they have tested negative in the initial screening?

a) What is the probability that selected person has the HIV virus if it is known that they tested positive in the initial screening?

b) What is the probability that selected person tests positive in the initial screening if is known that this person has the HIV virus?

c) What is the probability that selected person has the HIV virus if it is known that he tested negative in the initial screening?

d) What is the probability that selected person tests negative in the initial screening if it is known that he has the HIV virus?

To solve these probability questions, we can use the concept of conditional probability. Conditional probability is the probability of an event A occurring given that another event B has already occurred.

Let's define the events:
A: Selected person has the HIV virus
B: Selected person tests positive in the initial screening

Now let's solve each question step by step:

a) What is the probability that the selected person has the HIV virus if it is known that they tested positive in the initial screening?

To find this probability, we need to calculate P(A|B), which means the probability of having the HIV virus given that the person tested positive in the initial screening.

P(A|B) can be calculated using Bayes' theorem:

P(A|B) = (P(B|A) * P(A)) / P(B)

Here, P(B|A) is the probability of testing positive in the initial screening given that the person has the HIV virus. According to the information given, the preliminary screening test is correct 95% of the time, so P(B|A) = 0.95.

P(A) is the probability of a randomly selected person having the HIV virus, which is given as 10% or 0.10.

P(B) can be calculated using the law of total probability:
P(B) = (P(B|A) * P(A)) + (P(B|¬A) * P(¬A))

P(B|¬A) is the probability of testing positive in the initial screening given that the person does not have the HIV virus. Since the test is not perfect, there is a possibility of a false positive result. In this case, P(B|¬A) can be calculated as 1 - P(correct negative), where P(correct negative) is the probability of the test correctly identifying someone without the virus.
Since the test is known to be correct 95% of the time, P(correct negative) = 1 - 0.95 = 0.05.

P(¬A) is the probability of a randomly selected person not having the HIV virus, which is 90% or 0.90.

Now you can calculate P(B), and then substitute the values in Bayes' theorem to find P(A|B).

b) What is the probability that the selected person tests positive in the initial screening if it is known that they have the HIV virus?

To find this probability, we need to calculate P(B|A), which is the probability of testing positive in the initial screening given that the person has the HIV virus. This probability is given as 0.95.

c) What is the probability that the selected person has the HIV virus if it is known that they tested negative in the initial screening?

To find this probability, we need to calculate P(A|¬B), which is the probability of having the HIV virus given that the person tested negative in the initial screening.

P(A|¬B) can be calculated using Bayes' theorem:

P(A|¬B) = (P(¬B|A) * P(A)) / P(¬B)

P(¬B|A) is the probability of testing negative in the initial screening given that the person has the HIV virus. Since the test is not perfect, there is a possibility of a false negative result. In this case, P(¬B|A) can be calculated as 1 - P(correct positive), where P(correct positive) is the probability of the test correctly identifying someone with the virus.
Since the test is known to be correct 95% of the time, P(correct positive) = 0.95.

P(¬B) can be calculated using the law of total probability:
P(¬B) = (P(¬B|A) * P(A)) + (P(¬B|¬A) * P(¬A))

P(¬B|¬A) is the probability of testing negative in the initial screening given that the person does not have the HIV virus. This can be calculated as 1 - P(B|¬A), where P(B|¬A) is the probability of testing positive in the initial screening given that the person does not have the HIV virus. This probability was calculated in question (a).

Now you can calculate P(¬B), and then substitute the values in Bayes' theorem to find P(A|¬B).

d) What is the probability that the selected person tests negative in the initial screening if it is known that they have the HIV virus?

To find this probability, we need to calculate P(¬B|A), which is the probability of testing negative in the initial screening given that the person has the HIV virus. This probability is given as 1 - P(B|A), where P(B|A) is the probability of testing positive in the initial screening given that the person has the HIV virus. This probability was calculated in question (b).