A test for a certain disease is found to be 95% accurate, meaning that it will correctly diagnose the disease in 95 out 100 people who have the ailment. The test is also 95% accurate for a negative result, meaning that it will correctly exclude the disease in 95 our of 100 people who do not have the aliment. For a certain segment of the population the incidence of the disease is 4%.


(1) If a person tests positive, find the probability that the person actually has the disease (Hint: define appropriate events and use Bayes Theorem);

(2) Now suppose the incidence of the disease is 49%. Compute the probability that the person actually has the disease, given that the test is positive;

(3) The probability you obtained in (1) is much smaller than 0.95, if your computation is correct. Hence, You can conclude that there is only a much smaller probability to claim “the person really has the disease” after knowing that “the test is positive”, though the test has 95% “correctness”. Explain this difference.
\\Hint.
Population: the segment of the population
Experiment: Determine whether or not a selected person has the disease
Outcomes: D, Dc (where D and Dc are defined below)
Sample space: S = {D, Dc}
Define necessary events: D = the selected person has the disease;
Dc = the selected person does not have the disease.
Rate of disease = 4% ? P(D) = 4% , P(Dc) = 96%
E = Testing result shows positive for the selected person
Ec = Testing result shows negative for the selected person
Correctness of testing: if the selected person does have the disease, the testing will show positive
with probability 95%, i.e., (about) 95 positive out of 100 persons who do have the disease.
if the selected person does NOT have the disease, the testing will show negative with
probability 95%, i.e., (about) 95 positive out of 100 persons who do NOT have the disease.
In other words, P(E | D) = 95% and P(Ec | Dc) = 95%
Want P(D | E) = ?



***Please help I have a test to do and this is an example of what I have on my test, if someone could solve this for me, and please work it out so I will know how to do it step by step plz\n