Can someone please help me? certain medical diagnostic test is used to determine if people have pinkyitis and it is known that 1 in 1300 Americans has the disease. The test correctly diagnoses the presence of pinkyitis 97% of the time and it correctly diagnoses the absence of the disease 98% of the time. Find the probability that a person has the disease given that the test says that he has the disease. Find the probability that a person does not have the disease given that the test says that he does not have the disease

To find the probability that a person has the disease given that the test says he has the disease, we can use Bayes' theorem.

Let's denote:
A - the event that a person has the disease
B - the event that the test says the person has the disease

We know that the probability of having the disease is given as 1 in 1300, which can also be written as P(A) = 1/1300.

We also know that the test correctly diagnoses the presence of the disease 97% of the time, which means it has a 97% accuracy rate, or P(B|A) = 0.97.

Bayes' theorem states:
P(A|B) = (P(B|A) * P(A)) / P(B)

To find P(B), the probability that the test says a person has the disease, we need to consider both scenarios: when the person actually has the disease (true positive) and when the person doesn't have the disease but the test says they do (false positive).

P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)

Given that the test correctly diagnoses the absence of the disease 98% of the time (2% false positive rate), we have P(B|not A) = 0.02. The complement of having the disease, P(not A), can be calculated as 1 - P(A).

Evaluating the above equation, we can find P(B).

Next, we can substitute the values into Bayes' theorem to find P(A|B), the probability that a given person has the disease given that the test says they have the disease.

To find the probability that a person does not have the disease given that the test says they do not have the disease, we can use the same approach with different values.

Let's denote:
C - the event that a person does not have the disease
D - the event that the test says the person does not have the disease

In this case, we know that P(C|D) is the probability we want to find. Using Bayes' theorem again, we can calculate it by following the same steps as above with the given values of P(D|C) and P(C).