A certain disease is difficult to be diagnosed and the probability of correctly diagnosing the disease is 0.6. If any patient, after the correct diagnosis, has 40% chances of dying. However, an incorrect diagnosis enhances the probability of death to 0.7. If a patient has died after the treatment, what is the probability that the disease was diagnosed correctly?
6/13
This is a conditional probability problem. Using Bayes’ theorem, we calculate the probability that the diagnosis was correct given that the patient died.
P(Correct | Died) = (P(Died | Correct) * P(Correct)) / (P(Died))
P(Died | Correct) = 0.4, P(Correct) = 0.6, P(Died) = P(Died | Correct) * P(Correct) + P(Died | Incorrect) * P(Incorrect)
P(Died) = 0.4 * 0.6 + 0.7 * 0.4 = 0.24 + 0.28 = 0.52
So, P(Correct | Died) = (0.4 * 0.6) / 0.52 ≈ 6/13.
