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?
$$\mathsf P(N_e=10\mid N_e+N_p=90) = \binom{90}{10} 0.10^{10}0.90^{80}$$
I suspect that you have your notes confused.