Suppose I know that events occur according to a splitting Poisson process of rate $\lambda = 10$ per hour. Suppose further that events is categorized as type $1$ with probability $\frac{1}{4}$ and as type $2$ with probability $\frac{3}{4}$. Assume that a total of $200$ events arrived in the first $10$ hours. Conditional on this, I would like to find the probability that $n$ events are categorized as type $1$ in the first $4$ hours.
I know that $P(X(4) = N|X(10)=200) = \dbinom{200}{N}\left(\frac4{10}\right)^N\left(\frac6{10}\right)^{200-N}$. Hence, given $200$ events arrived in $10$ hours, the number of events occurring in $4$ hours is distributed $Bin\left(N,\frac{4}{10}\right)$. Now, conditional on $N$ events occurring in $4$ hours, the number of events categorized as being of category $1$ is binomially distributed. If I let $N_1(4)$ denote the number of events being categorized as category $1$ in $4$ hours, then $N_1(4)$ conditional on $X(4) = N$ is $Bin\left(N, \frac{1}{4}\right)$.
I understand that conditional binomial events are conditional as well, and so conditional on $200$ events arriving in $10$ hours, the number of events arriving as category $1$ in $4$ hours from these $10$ hours is distributed $Bin\left(N, \frac{1}{4}\frac{4}{10}\right)$.
I am wondering how to express these in terms of probability notation. Does the following equations express what is going on correctly?
$$ P(N_1(4) = n_1|X(4) = N, X(10) = 200) = P(N_1(4)= n_1|X(4) = N) \cdot P(X(4) = N|X(10)=200) $$
where $P(N_1(4)= n_1|X(4) = N) \sim Bin\left(N, \frac{1}{4}\right)$
and $P(X(4) = N|X(10)=200) \sim Bin\left(N,\frac{4}{10}\right)$
and $P(N_1(4) = n_1|X(4) = N, X(10) = 200) \sim Bin\left(N, \frac{1}{4}\frac{4}{10}\right)$?