(16 total points) One of the goals of a Phase 1 vaccine trial is to determine the correct dosage. Often several doses are considered and the safety and effectiveness of each is measured. The chart contains hypothetical data on random variables X, the vaccine dosage (in micrograms) and Y, the presence of major side effects (Y=1 indicates there was a side effect, Y=0 indicates there wasn't).

a. (3 pts) What is the probability that a randomly selected patient that was given the 10 microgram dose had a major side effect?

b. (3 pts) What is the probability that a randomly selected patient had a major side effect and received the 3 microgram dose?

c. (3 pts) Calculate ( ).

d. (4 pts) Suppose the 10 microgram dose is chosen and the true probability of having a major side effect under that dose is equal to the value you calculated in part a. In the Phase II part of the trial, the vaccine is administered to a larger group of patients. Suppose the reactions of these patients are independent of each other. If a nurse administers 30 vaccines during his shift, what is the probability that exactly 4 patients have major side effects. e. (3pts) What is the expected number of patients the nurse would give the vaccine to until there are 2 major side effects?