MAT 120-A certain virus infects one in every 200 people.

1)A certain virus infects one in every 200 people. A test used to detect the virus in a person is positive 80% of the time if the person has the virus and 5% 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)Using bayes theorem when a person tests positive, determine the probability that a person is infected

b)Using bayes theorem when a person tests negative, determine the probability that a person is not infected

2) a scientist designed a medical test for a certain disease. Among 100 patients who have the disease, the test will show the presence of the disease in 97 cases out of 100 and will fail to show the presence of the disease in 3 cases out of 100. among those who do not have the disease, the test will erroneously show the presence of the disease in 4 cases out of 100 and will show there is no disease in the remaining 96 cases out of 100.

a)what is the probability that a patient who tested positive on this test actually has the disease, if it is estimated that 20% of the population has the disease

b)what is the probability that a patient who tested negative on this test actually does not have the disease if it is estimated that 4% of the population has the disease