It's only the last two parts of this I'm having a bit of bother with, but I'll type the whole question out so you know what's going on.
Suppose that the number of times during a year that an individual catches a cold can be modelled by a Poisson random variable with an expectation of 4. Further suppose that a new drug based on Vitamin C reduces the expectation to 2 (but is still a Poisson distribution) for 80% of the population, but has no effect on the remaining 20% of the population. Calculate
a) the probability that an individual taking the drug has 2 colds in a year if they are part of the population which benefits from the drug;
b) the probability that an individual has 2 colds in a year if they are part of the population which does not benefit from the drug;
c) the probability that a randomly chosen indicidual has 2 colds in a year if they take the drug;
d) the conditional probability that a randomly chosen individual is in the part of the population which benefits from the drug given that they had 2 colds in a year during which they took the drug.
Pretty lengthy, sorry.