Could someone explain this solution please?

[mewmew]

http://img478.imageshack.us/img478/5606/part1bd8.jpg
http://img183.imageshack.us/img183/686/part2yu0.jpg

We got this in class from a TA and the professor is in China and not able to answer questions. I am confused by where the probability,P(N), comes from in part a. It looks like a multiplicity multiplied by some other stuff but I don't understand it at all. I haven't had any probability/statistics but I assume it's pretty basic. If anyone could help me understand how this probability comes about it would be much appreciated. Thanks

 

[quasar987]

p=V/V0 is the probability that one molecule be in the volume V. q=1-p is the probability that it be somwhere else. This expression of q comes from the fact that since the particle must be somewhere we must have p+q=1.

Choose N particles among N0. The probability that these N be in V and that the rest of them are NOT in V is $p^Nq^{N_0-N}$. In general, the probability that exactly N particles be in V and the rest not in V is $p^Nq^{N_0-N}$ summed over as many ways there are to choose which N particles among N0 are going to be in V. And you probably at least know some basic probability results, among which that the number of ways to chose N amongst N0 is $$\binom{N_0}{N}=\frac{N_0!}{N!(N_0-N)!}$$

So we have the result.

 

[mewmew]

Thanks, that makes it much clearer.