I'm having difficulties with this problem:
Suppose you have an entire city afflicted with four distinct and exclusive diseases and a laboratory is assigned to test which disease each citizen has.
The reliability of these tests are as follows:
Disease A = 72.7%
Disease B = 81.1%
Disease C = 75.2%
Disease D = 80.1%
The percentage of the population of people afflicted is as follows:
P(B1) = 18.1% (Disease A)
P(B2) = 31.9% (Disease B)
P(B3) = 18.9% (Disease C)
P(B4) = 31.1% (Disease D)
If a random person were to selected from the entire population and then tested positive for disease A, what is the probability that they actually have disease A?
I think that the problem is asking for P(B1|A1), so I used this formula:
P(B1|A) = P(A|B1)P(B1) / ( P(A|B1)P(B1) + P(A|B2)P(B2) + P(A|B3)P(B3) + P(A|B4)P(B4) )
These are the values that I am sure of:
P(A|B1) = .727, because that is the chance of a true positive result of disease A being detected
P(B1) to P(B4) = the population listed above, corresponding to A, B, C and D.
The problem is now, I don't know what values to put inside P(A|B2) to P(A|B4)
Do I put in just the rate of false positive (.273)? Or do I use the corresponding tests for disease B, C and D (.811, .752, .801, respectively)? Or am I missing something here?