Probability of finding $n_1$ particles in volume $v_1$ and $n_2$ particles in volume $v_2$

Question

In these lecture notes (http://cgarrod.org/Book/chapter%201.pdf), they explain how to compute the probability of finding $n$ indistinguishable particles in volume $v$, given that there are $N$ particles and the total volume is $V$. The formula is: $$ P[\text{$n$ particles in $V$}] = \binom{N}{n} p^n (1-p)^{N-n} $$ where $p$ is $$ p = \frac{v}{V} $$

I would like to know how this formula generalizes if one wants to compute "the probability of finding $n_1$ particles in volume $v_1$ and $n_2$ particles in volume $v_2$, given that there are $N$ indistinguishable particles in total, that the total volume is $V$ and that $v_1$ and $v_2$ are disjoint."

Answer 1

If you consider $p_1=v_1/V$ and $p_2=v_2/V$ then $$P[n_1~\mbox{particles in}~v_1~\mbox{and}~n_2~\mbox{particles in}~v_2]=\frac{N!}{n_1!n_2!(N-n_1-n_2)!}p_1^{n_1}p_2^{n_2}(1-p_1-p_2)^{N-n_1-n_2}$$ That's because if you choose $n_1$ particles, the probability they all fall into $v_1$ is $p_1^{n_1}$, then, if you choose $n_2$ other particles, the probability they all fall into $v_2$ is $p_2^{n_2}$ and the probability the remaining $N-n_1-n_2$ particles doesn't fall into $v_1$ or $v_2$ is $(1-p_1-p_2)^{N-n_1-n_2}$. Finally, there are $\frac{N!}{n_1!n_2!(N-n_1-n_2)!}$ ways of choosing the particles that way as explained in (1.24) in the lecture notes you linked.

This is called a multinomial distribution and it can be generalized to $k$ disjoint volumes.