Probability of finding n 1 particles in volume v 1 and n 2 particles in volume v 2 .The formula is: P [ n particles in V ] = ( N n ) p n ( 1 − p ) N − n where p is p = 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."

Question

Probability of finding       n    1   particles in volume       v    1   and       n    2   particles in volume       v    2  .The formula is:  P  [            n         particles in           V        ]  =                    (                    N      n                      )                  p    n    (  1  −  p      )          N      −      n       where p is   p  =      v    V  .I would like to know how this formula generalizes if one wants to compute &quot;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.&quot;

encaselatqr · Accepted Answer

Step 1If you consider       p    1    =      v    1        /    V and       p    2    =      v    2        /    V then   P  [      n    1           particles in           v    1           and           n    2           particles in           v    2    ]  =            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            Step 2That&#039;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&#039;t fall into       v    1   or       v    2   is   (  1  −      p    1    −      p    2        )          N      −              n        1            −              n        2            . Finally, there are             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.

Probability of finding n_{1} particles in volume v_{1} and n_{2} particles in volume v_{2}.

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