Suppose we've an isolated box having $N$ classical distinguishable particles in it, the box being hypothetically divided into two parts, left and right with both parts identical.

Reif says

...Note that there is only one way of distributing the $N$ molecules so that all $N$ of them are in the left half of the box. It represents only one special configuration of the molecules compared to the $2^{N}$ possible configurations of these molecules. Hence we would expect that, among a very large number of frames of the film, on the average only one out of every $2^{\mathrm{N}}$ frames would show all the molecules to be in the left half. If $P_{N}$ denotes the fraction of frames showing all the $N$ molecules located in the left half of the box, i.e., if $P_{N}$ denotes the relative frequency, or probability, of finding all the $N$ molecules in the left half, then $$ P_{N}=\frac{1}{2^N} . $$

Why should we expect that, among a very large number of frames of the film, on the average only one out of every $2^{\mathrm{N}}$ frames would show all the molecules to be in the left half.

Suppose we've an isolated box having $N$ classical distinguishable particles in it, the box being hypothetically divided into two parts, left and right with both parts identical.

Reif says

...Note that there is only one way of distributing the $N$ molecules so that all $N$ of them are in the left half of the box. It represents only one special configuration of the molecules compared to the $2^{N}$ possible configurations of these molecules. Hence we would expect that, among a very large number of frames of the film, on the average only one out of every $2^{\mathrm{N}}$ frames would show all the molecules to be in the left half. If $P_{N}$ denotes the fraction of frames showing all the $N$ molecules located in the left half of the box, i.e., if $P_{N}$ denotes the relative frequency, or probability, of finding all the $N$ molecules in the left half, then $$ P_{N}=\frac{1}{2^N} . $$

Why should we expect that, among a very large number of frames of the film, on the average only one out of every $2^{\mathrm{N}}$ frames would show all the molecules to be in the left half?

I have the intuition for it but how can one show this should be the case indeed rigorously?

Why can't the average be more or less than the above value? Also how large is "very large frames of film" here?

Suppose we've an isolated box having $N$ classical distinguishable particles in it, the box being hypothetically divided into two parts, left and right with both parts identical.

Reif says

...Note that there is only one way of distributing the $N$ molecules so that all $N$ of them are in the left half of the box. It represents only one special configuration of the molecules compared to the $2^{N}$ possible configurations of these molecules. Hence we would expect that, among a very large number of frames of the film, on the average only one out of every $2^{\mathrm{N}}$ frames would show all the molecules to be in the left half. If $P_{N}$ denotes the fraction of frames showing all the $N$ molecules located in the left half of the box, i.e., if $P_{N}$ denotes the relative frequency, or probability, of finding all the $N$ molecules in the left half, then $$ P_{N}=\frac{1}{2 N} . $$

Why should we expect that, among a very large number of frames of the film, on the average only one out of every $2^{\mathrm{N}}$ frames would show all the molecules to be in the left half.

Suppose we've an isolated box having $N$ classical distinguishable particles in it, the box being hypothetically divided into two parts, left and right with both parts identical.

Reif says

...Note that there is only one way of distributing the $N$ molecules so that all $N$ of them are in the left half of the box. It represents only one special configuration of the molecules compared to the $2^{N}$ possible configurations of these molecules. Hence we would expect that, among a very large number of frames of the film, on the average only one out of every $2^{\mathrm{N}}$ frames would show all the molecules to be in the left half. If $P_{N}$ denotes the fraction of frames showing all the $N$ molecules located in the left half of the box, i.e., if $P_{N}$ denotes the relative frequency, or probability, of finding all the $N$ molecules in the left half, then $$ P_{N}=\frac{1}{2^N} . $$

Why should we expect that, among a very large number of frames of the film, on the average only one out of every $2^{\mathrm{N}}$ frames would show all the molecules to be in the left half.