A new treatment for a disease is being tested, to see whether it
is better than the standard treatment. The existing treatment is
effective on 50% of patients. It is believed initially that there
is a 2/3 chance that the new treatment is effective on 60% of
patients, and a 1/3 chance that the new treatment is effective on
50% of patients. In a pilot study, the new treatment is given to 20
random patients, and is effective 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. the probability that the new treatment is
better than the standard treatment is p , what is the PMF for how
many of the new patients the new treatment is effective
on?