Question: A new treatment for a disease is being tested, to see whether it is better than the standard treatment. The existing treatment is defective on 50% of patients. It is believed initially that there is a 2/3 chance that the new treatment is defective on 60% of patients, and a 1/3 chance that the new treatment is defective on 50% of patients. In a pilot study, the new treatment is given to 20 random patients, and is defective for 15 of them.

(a) Given this information, what is the probability that the new treatment is better than the standard treatment?

(b) A second study is done later, giving the new treatment to 20 new random patients. Given the results of the first study, what is the PMF for how many of the new patients the new treatment is e↵ective on? (Letting p be the answer to (a), your answer can be left in terms of p.)