Let event A1 : Disease in serious form,
event A2 : Disease in mild form,
event A3 : Subject does not have disease,
event B: Subject tests positive.
P(A1 ) = 0.02, P(A2) = 0.1, P(A3) = 0.88
The probability of testing positive is 9/10 if the subject has the serious form, 6/10 if the subject has the mild form, and 1/10 if the subject doesn’t have the disease.
∴ P(B/A1) = 0.9, P(B/A2) = 0.6, P(B/A3) = 0.1
P(B) = P(A1) P(B/A1) + P(A2) P(B/A2 ) + P(A3) P(B/A3)
= 0.02 × 0.9 + 0.1 × 0.6 + 0.88 × 0.1
= 0.166
Required probability = P(A1 /B)
By Baye’s theorem