Poisson process conditional probability problem 
Penguins slide through a chute in a Poisson process at a rate of $2$ per minute. Each penguin has a $10$% chance of being an emperor penguin, independent of everything else. Given that $90$ penguins shot through the chute in an hour, what is the probability that $10$ were emperors?

I haven't come across a Poisson process question like this before, but figured that I should probably make use of the Binomial distribution, making use of the fact that given $X(t) = n$ for a Poisson process, then $X(s) \sim \text{Bin}(n,\frac{s}{t})$ when $s \leq t$. 
So in this case, the probability that $10$ were emperors would follow a Binomial distribution with $n=90$ and $\frac{s}{t} = 0.1$. That would give an answer of
$${90 \choose{10}} (0.1)^{10} (0.9)^{80} = 0.125
$$

Now a few problems. 


*

*I got this question marked wrong on my assignment, so I must have the wrong answer! However I cannot figure out how my answer could be wrong, and in what other manner I could approach this question. 

*I don't really understand the notation used in the idea used in my own approach (from my notes). For $X(s) \sim \text{Bin}(n,\frac{s}{t})$, does $s$ represent the time period we are covering, which in this case is equivalent to $t$? If so, I don't really understand what $s/t$ is useful for in a Binomial distribution. In this case I sort of blindly followed a break in logic and used a Binomial with $0.10$ as my probability value because intuitively it made sense to me. I got part marks but no comments on my approach, so I'm unsure if they're purely out of sympathy or if I was on the right track?
 A: Your answer is ok except that the parts about $s$ and $t$ seems confused.  Only one time period is mentioned, and how much time it is doesn't matter because we are given the number of arrivals during that time. (If we were asked "What is the probability that the number of arrivals will be $90$?" that would be another matter.)  No ratio of times is involved.  The number $0.1$, the probability of each penguin's being an emperor penguin, does not come from a ratio of times.
Here's a guess: If the question is "What is the probability that the number of penguins in the first $24$ minutes was $30$ given that the number in the first hour is $90$?", then it would be correct to say that the number in the first $24$ minutes is distributed as $\mathrm{Bin}(90,24/60)$.  However, that is not what the question was.
A: *

*Yes, the conditional distribution of the count of emperor penguins among the given total of $90$ penguins will be binomial with parameters $n=90, p=0.10$.   The fact that the penguins arrive in a Poisson process is irrelevant to the question.


$$\mathsf P(N_e=10\mid N_e+N_p=90) = \binom{90}{10} 0.10^{10}0.90^{80}$$


*However, the notations for random variables $X(s), X(t)$ is more suggestive of count of penguins arriving in period lengths $s, t$.   This has nothing to do with the question.


I suspect that you have your notes confused. 
