Probability density of a 1-D Tonk Gas

[GravityX]

Homework Statement

Show that the probability density of an arbitrary $y_i$ is: $p(y_i=y)=\frac{N(L_f-y)^{N-1}}{L_f^N}$ for $0\leq y \leq L_f$

Relevant Equations

none

It is a 1D Tonk gas consisting of $N$ particles lined up on the interval $L$. The particles themselves have the length $a$. Between two particles there is a gap of length $y_i$. $L_f$ is the free length, i.e. $L_f=L-Na$.

I have now received the following tip:

Determine the relative size of the slice through location space defined by a given $y_1$. Visualize the case $N=2$.

Is the following meant by relative size? $\frac{y_1}{L}$

Unfortunately, I can't do anything with the tip because I don't know what exactly I have to do.

 

[haruspex]

Homework Statement:: Show that the probability density of an arbitrary $y_i$ is: $p(y_i=y)=\frac{N(L_f-y)^{N-1}}{L_f^N}$ for $0\leq y \leq L_f$
Relevant Equations:: none

It is a 1D Tonk gas consisting of $N$ particles lined up on the interval $L$. The particles themselves have the length $a$. Between two particles there is a gap of length $y_i$. $L_f$ is the free length, i.e. $L_f=L-Na$.

I have now received the following tip:

Determine the relative size of the slice through location space defined by a given $y_1$. Visualize the case $N=2$.

Is the following meant by relative size? $\frac{y_1}{L}$

Unfortunately, I can't do anything with the tip because I don't know what exactly I have to do.

My guess is that location space means an N-dimensional cube of side L_f. The locations of the particles are then representable by a point in the cube.
For two particles, you have a square. The positions of the particles, measured from one end, are x, y. By choosing y>x, you have only a triangle to consider, and their separation is y-x. So in the triangle, fix the value of y-x and determine the line of points (x,y) which satisfy that. How long is the line, as a function of y-x?