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 = ( ρ 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.I know the probability P ( n 1 , n 2 , … , n M ) of finding the system in a particular configuration in the observation volume is p ( N ) = p N ( 1 − p ) M − N ,, (Bernoulli Distribution), and since there are N particles in M spaces then the maximum number of configurations is M ! ( N ! ( M − N ) ! ) I'm not sure where to go from here.

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Probability of Observing N particles in a given volume?I&#039;m having an issue with a probability problem concerning solutions.Assume there is an &quot;observational region&quot; 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  =  (            ρ        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.I know the probability   P  (      n    1    ,      n    2    ,  …  ,      n    M    ) of finding the system in a particular configuration in the observation volume is   p  (  N  )  =      p    N    (  1  −  p      )          M      −      N        ,, (Bernoulli Distribution), and since there are N particles in M spaces then the maximum number of configurations is             M      !              (      N      !      (      M      −      N      )      !      )      I&#039;m not sure where to go from here.

Nathalie Rivers · Accepted Answer

Step 1If 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  (  N  )  =  p  (  N  )  W  (  N  )  =      p    N    (  1  −  p      )          M      −      N        W  (  N  )  ..Step 2Every 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).

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