The Question:
The number of people entering a restaurant is a random variable $N$ with poisson distribution with parameter $\lambda=36$.
every one of the people enjoys his/her meal with a probability of $\frac{1}{3}$ independently from the number of people that entered the restaurant and other people's decisions (enjoyed or not).
Find the probability function of the number of people that enjoyed their meal given that the number of people entered the restaurant is $n$. in other words, if $E$ is a random variable that counts how many people enjoyed their meal, then find $P(G=k|N=n)$.
The Way I solved it:
Since it's said that the people which enjoy their meal are independent of the number of people that entered the restaurant, what I did is simply: $P(G=k|N=n)=P(G=k)$ , and I know that $G\sim Pois(\frac{1}{3}*36)$.
The Answer:
They introduced $I_i$ indicator random variable that is equal to $1$ whenever person $i$ enjoyed his meal and $0$ when not.
And so $$P(G=k|N=n)=P(\sum_{j=0}^{N}I_i=k | N=n)=P(\sum_{j=0}^{n}I_i=k | N=n)=P(\sum_{j=0}^{n}I_i=k)={n \choose k}(\frac{1}{3})^k(\frac{2}{3})^{n-k}$$
Notice that they did also use the independence in step 3.
My Confusion and my thoughts:
I completely understand the idea behind the answer, and intuitively I understand why it's correct, as we're setting the number of trials to $n$, and so it's no longer infinite, and thus it turns into a binomial distribution whether each person enjoyed their meal or not with $n$ trials.
What confuses me is wheres my mistake? why when I used the independence at the first step it completely gave me a wrong answer, I know that it made me lose the given information that $N=n$, but it was said that they're not dependent, so why did I get this outcome?
Any help in clearing this confusion is really appreciated, thanks in advance!
The question says that the probability of a person enjoying their meal is independent of the number of people in the restaurant. It does not say that the number of people enjoying their meal is independent of the number of people in the restaurant.