A certain virus infects 10 in every 1000 people.

A test used to detect the virus in a person is positive 80% of the time when the person has the virus (20% false negative), and 5% of the time when the person does not have the virus (5% false positive).

That is 20% of the time when the test should be positive but didn't, and 5% of the time the test indicated positive and shouldn't.

1) Out of all people who tested positive, what portion is really infected? In another word, if a person is tested positive, what is the probability that the person is infected by the virus?

2) If the infection rate for the population is 50 in every 1000 people, then when a person is tested positive what is the probability that the person is really infected by the virus?

3) If we are able to improve the accuracy, the false negative rate was reduced to 10% and the false positive rate was reduced to 3%. And the population infection rate remains at 10 in 1000 people. Under this scenario, if a person is tested positive what is the probability that the person is really infected by the virus?

4) Use your answer from part 1) as the baseline, compare your answers from 2) and 3) with the baseline. How did the probability change (increase or decrease) when the population infection rate increased from 10 in 1000 to 50 in 1000? How did the probability change (increase or decrease) when the test accuracy was improved?