Problem 2 (10 points) Assume a doctor knows that a patient is sick either with disease D₁ or with disease D₂, and the two events are taken as the complement of each other. It is also known that P(D₁) = 0.9. A symptom S may happen with both diseases with P(S|D₁) = 0.4 and P(S|D₂) = 0.8.

1. If the patient shows S, what is the probability that the patient has D₂?

1. If the patient does not show S, what is the probability that the patient has D₂?