Math Help - Poisson distribution

1. Poisson distribution

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.

2. Originally Posted by chella182
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.
I've seen longer, and the context is never bad to have, so thanks!

c) hint: $\mathbf{P}(N=2) = \mathbf{P}((N=2 \textrm{ and they benefit}) \textrm{ or } (N=2 \textrm{ and they do not benefit}))$

break out "or", note the two sets are disjoint so the intersection is empty
$= \mathbf{P}(N=2 \textrm{ and they benefit}) + \mathbf{P}(N=2 \textrm{ and they do not benefit})$ - 0

$= \mathbf{P}(N=2 |\textrm{they benefit})\mathbf{P}(\textrm{they benefit}) + \ldots$

d) $\mathbf{P}(\textrm{they benfit}|N=2) = \mathbf{P}(\textrm{they benefit and }N=2)/\mathbf{P}(N=2)$

You had 2 figure out what both the numerator and denominator were in the previous part - if you followed my hint

3. Thanking you muchly looking at my answers for part a) and b) I'm not sure they're right, but still haha.

4. Poisson distribution problem

Okay, so I've looked at my answers to parts a) and b) to the question in the first post & I don't think they're right. Can anyone help?