Clarification on notation from a probabilty/quantum mechanics question

Question

I'm working through Stephen Barnett's book on quantum information and have come across the following question (1.5, for anyone keeping track at home)

A particle counter records counts with an efficiency $\eta$. This means that each particle is detected with probability $\eta$ and missed with probability $1-\eta$. Let $N$ be the number of particles present and $n$ be the number detected. Show that

$$P(n|N) = \frac{N!}{(N-n)!n!}\eta^n(1-\eta)^{N-n}$$ [I did this part, no problem]

a) Calculate P(N|n) for

$$P(N) = e^{-\bar{N}}\frac{\bar{N}^N}{N!}$$

b) Calculate P(N|n) for all P(N) equally probable

c) Calculate P(N|n) given only that the mean number of particles present is $\bar{N}$.

My questions:

1. What does $P(N)$ denote? I recognize that question $a)$ is the Poisson distribution, so I assume that it's the probability of $N$ particles being detected, given a mean of $\bar{N}$. But this conflicts with the information in the first paragraph... It seems like $N$ is being used as a constant in the first paragraph, and now as a variable in the question.

2. Does $P(N|n)$ mean "the probability that $N$ particles are detected, given that $n$ are"?

3. Doesn't question $c)$ conflict with the fact that the first paragraph says that $N$ particles are present?

Answer 1

1. For ease of notation, let's relabel so that $Q$ is the number of particles present and $R$ is the number of particles observed. Then they are asking for $P(Q=q \mid R=r)$, which is $\frac{P(Q=q \wedge R=r)}{P(R=r)}=\frac{P(Q=q \wedge R=r)}{\sum_{s=r}^\infty P(Q=s \wedge R=r)}$.
2. Sticking to the above notation, the meaning of $P(Q=q \mid R=r)$ is "the probability that there are $q$ particles present given that $r$ particles were observed". This is really the quantity that you would care about if you were trying to estimate how many particles were there by using an error-prone detector. Ideally it would be nearly $1$ for $q=r$ and nearly zero for all other values of $q$ with fixed $r$.
3. All three of a,b,c are dropping the assumption that $N$ is a given number, and setting it to be a random variable instead. (However, I don't really know what the problem expects you to do in parts b and c.)