1. Prove that if A and B are independent so are
(a) A and B˜.
(b) A˜ and B˜.
2. A doctor assumes that a patient has one of three diseases d1, d2, or d3. Before any test, he assumes an equal probability for each disease. He carries out a test that will be positive with probability .8 if the patient has d1, .6 if he has disease d2, and .4 if he has disease d3. Given that the outcome of the test was positive, what probabilities should the doctor now assign to the three possible diseases?