Question 1 (22 points) A rapid antigen test for Covid-19 has the following characteristics. If an individual has an active Covid-19 infection, the test comes back positive 94% of the time. If an individual does not have an active Covid-19 infection, the test comes back negative 98% of the time. In a particular city, it is believed that 1% of all individuals currently have an active Covid-19 infection.

a) What is the conditional probability that an individual from the city tests positive, given that they do not have an active Covid-19 infection? (4 points)

b) Suppose that an individual from the city takes the test once, and tests positive. What is the probability that the individual has an active Covid-19 infection?

c) Are the events "individual tests positive" and "individual does not have an active Covid-19 infection" collectively exhaustive? (Select only one answer below. You do not need to show your work for this question.) (3 points)

i. Not enough information to determine.

ii. No, because it is possible for an individual to test positive and not have an active Covid-19 infection.

iii. No, because some individuals test negative and do have an active Covid-19 infection.

iv. Yes, because some individuals who test positive have an active Covid-19 infection, and some individuals who do not have an active Covid-19 infection test negative.

v. Yes, because it is possible for an individual to test positive and not have an active Covid-19 infection.

For parts d) and e) below, you may assume that the following is additionally known about the test when the test is administered more than once. If an individual has an active Covid-19 infection, then each test that the individual takes is independent of the others that the individual takes. Similarly, if the individual does not have an active Covid-19 infection, then each test that the individual takes is independent of the others that the individual takes.

d) Suppose that an individual from the city intends to take the test four times. What is the probability that tests 1 and 2 come back positive, and tests 3 and 4 come back negative? Give your answer to 6 decimal place.

e) An outspoken tech company CEO living in the city decides to take the test. Tests 1 and 2 come back positive, while tests 3 and 4 come back negative. What is the probability that this CEO has an active Covid-19 infection?