They are asking for: .
What the probability that the paitent is ill and the test shows ill?
Using Bayes' Theorem on : .
Substitute : .
A= the paitent is ill: P(A)=0,001
A*= the paitent is not ill
B= the diagnos shows ill
B*=the diagnos shows not ill
given: p(B|A)= 0,8
Question: What the probability that the paitent is ill and the diagnos shows ill?
started with: P(A|B)= P(A∩B)/P(B)= (P(A)xP(B|A))/P(B)