Virus test:- A certain exotic virus cannot be detected with certainty, but a simple saliva bleach test is available that is somewhat reliable. If the virus is present, the test gives a positive result 60 percent of the time and a negative result 40 percent of the time. If the virus is not present, the test gives a (false) positive result 20 percent of the time. The test is only administered to people who have symptoms characteristic of the virus infection, and it is known that 30 percent of the people with these symptoms actually have the virus.
(a) What is the probability that someone with the symptoms will have a positive test result?
(b) What is the mutual information I(A; B) where A is the state of the virus, and B is the test result?
(c) Suppose that if you have the virus and are treated, you recover 100 percent of the time. However, it is not a comfortable treatment and you will have to terminate your vacation. The treatment has a negative value to you whether or not you have the virus. On the other hand, not treating the virus will mean a difficult time, which has a much greater negative value. You might construct the payoff structure shown in the table below.
(d) Find the value V∗(A|B), which is the value with the test information.
(e) Find the value of information.