# Probabilities and binomial theorem

1. Sep 27, 2011

### SoggyBottoms

1. The problem statement, all variables and given/known data

Consider an ideal gas of N identical particles in a volume V, and a subvolume v. The chance a molecule is in inside the subvolume is P = v/V.

a) What is the chance the subvolume contains n particles?

b) Use the binomial theorem $$(p + q)^N = \sum_{n = 0}^N p^n q^{N-n} \frac{N!}{n!(N-n)!}$$ for random p and q to calculate the average number of particles $$\langle n\rangle$$ and the variance $$\sigma^2 = \langle n^2 \rangle - \langle n \rangle^2$$

Anyone who could give me a nudge in the right direction?

2. Sep 27, 2011

### Eynstone

As the gas is ideal, the particles can go inside the subvolume independently. Thus, the answer is (v/V)^N =p^N. Part (b) follows from straightforward but long calculations.