7. (Poisson Distribution.) Let’s calculate the probability of having n particles in a subvolume V, for a box with total volume KV and a total number of particles T = KN0. (a) Find the exact formula for this probability: n particles in V, with total of T particles in KV. [Hint] What is the probability that the first n particles fall in the subvolume V, and the remainder T - n fall outside the subvolume (K - 1)V? How many ways are there to pick n particles from T total particles? (b) As K → ∞, show that the probability that n particles fall in the subvolume V has the Poisson distribution. What is the mean of this distribution? [Hint] You’ll need to use the fact that e -a = (e -1/K)Ka → (1 - 1/K) Ka as K → ∞, and the fact that n ≪ T. Here don’t assume that n is large: the Poisson distribution is valid even if there are only a few events.