Assume there is an "observational region" in a dilute solution with a volume V, and as solutes move across its boundary, the number N of solute molecules inside the observation region fluctuates. Divide V into M regions of volume v each with n particles. The solution is dilute enough that n=0 or 1 (there is no v with more than one particle of solute), and each cell is occupied (n=1) with probability rho=(rho_0)v. If W(N) is the number of configurations of the observation volume when N solutes are present, what is the probability P(N) of observing a given value of N, in terms of p, W(N), M, and N.

Probability of Observing N particles in a given volume?
I'm having an issue with a probability problem concerning solutions.
Assume there is an "observational region" in a dilute solution with a volume V, and as solutes move across its boundary, the number N of solute molecules inside the observation region fluctuates.
Divide V into M regions of volume v each with n particles. The solution is dilute enough that $n=0$ or 1 (there is no v with more than one particle of solute), and each cell is occupied ($n=1$) with probability $p=\left({\rho }_{0}\right)v$.
If W(N) is the number of configurations of the observation volume when N solutes are present, what is the probability P(N) of observing a given value of N, in terms of p,W(N),M, and N.
I know the probability $P\left({n}_{1},{n}_{2},\dots ,{n}_{M}\right)$ of finding the system in a particular configuration in the observation volume is $p\left(N\right)={p}^{N}\left(1-p{\right)}^{M-N},$, (Bernoulli Distribution), and since there are N particles in M spaces then the maximum number of configurations is $\frac{M!}{\left(N!\left(M-N\right)!\right)}$
I'm not sure where to go from here.
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Nathalie Rivers
Step 1
If p(N) is the probability of finding a given configuration of N particles, and W(N) is the amount of configurations for a given N number of particles, then the probability of observing N particles total is the sum of the probability of finding each of the configurations that have N particles. That is, $P\left(N\right)=p\left(N\right)W\left(N\right)={p}^{N}\left(1-p{\right)}^{M-N}W\left(N\right).$.
Step 2
Every possible configuration has the same probability of occurring, so you just count how many of those meet your requisites (i.e., that they have N particles).