1.

A vessel of capacity V_0 contains n molecules. Calculate the probability of all the molecules assembling in a part of the vessel V lt V_0.

Answer»


Solution :First method. Since both parts of the vessel are equivalent (this is the consequence of the homogeneity and the isotropy of space) the probability of finding a particle INSIDE a VOLUME `V lt V_0` is equal to the ratio of volumes:
`p = V/(V_0) , q = 1 - p = (V_0 - V)/(V_0)`

In the present problem k = n, i.e. all the molecules are contained inside V. The answer is obtained from the solution of Problem 18.5. For a large number of molecules the probability of such an event is negligible.
Second method. The problem may be solved without the binomial distribution, bul directly on the basis of the THEOREM for compound probability. Spocificully, the probability of finding a molecule inside the given volume is `p = V//V_0)`. The probability of finding simultaneously all the n molecules in it is the PRODUCT of the INDIVIDUAL probabilities : `w = p^n = (V//V_0)^n`


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