7. (20 points) Your doctor is testing you for a disease. Exactly 1/5 of people in your area have this disease. The doctor offers you a choice between two tests, Test 1 and Test 2. These tests have the following properties: 0 Test 1: if you are sick, test 1 will be positive with probability 3 /4, and if you are not sick, test 1 will be positive with probability 1 / 4. 0 Test 2: if you are sick, test 2 will be positive with probability 1, and if you are not sick, test 2 will be positive with probability 1/2. (a) What is the probability that Test 1 will come back positive? (Hint: use the law of total probability.) (b) What is the probability that Test 2 will come back positive? (0) According to the principle that the best observation to make is the one whose outcome has the greatest entropy, which test should you take? (d) Let s be the outcome that you are sick, and +1 the outcome that test 1 comes back positive. Find the pointwise mutual information pmi(s, +1) of these two outcomes