Ex: The chances that doctor A will diagnose X correctly is 60%. The chances that a patient will die by his treatment after correct diagnosis is 40% and the chance of death by wrong diagnosis is 70%. A patient of doctor A, who had disease X, died. What is the chance that his disease was diagnosed correctly?
Solution: Let us define the following events:
E1 : Disease X is diagnosed correctly by doctor A.
E2: Disease X is not diagnosed correctly by doctor A.
E : A patient (of Dr. A), who had disease X dies.
Then we are given:
P (E1 ) = 0.6 P (E│E1) = 0.4
P (E2 ) = 1- P (E1) = 1-0.6 = 0.4 P (E│E2) = 0.7
Therefore, P (E) = P (E1). P (E│E1 ) + P (E2). P(E│E2) (Died of correct and wrong diagnosis). P(E∩E1) + P (E∩E2)
= 0.6 * 0.4 + 0.4* 0.7
= 0.24+0.28
= 0.52
Using Bayes’ theorem, the required probability is given by:
P(E1). P(E│E1)
P (E1│E) = ——————
P (E)
0.6*0.4
= ———- = 6/13.
0.52
