If two points are chosen at random from the Cantor set, what is the average distance between them? I've so far figured that I need to look at different cases for if the points are within the lower or upper third of the Cantor set, but am confused on how to proceed beyond that to take an average for this set.
I found this solution, but didn't quite understand it: http://people.missouristate.edu/lesreid/AdvSol37.html?
All help is appreciated.
 A: Here is the explanation; Let me call the steps of construction the Cantor set as
$S_i\,\, (i=0,1,2,...),$
where
$S_0 = [0,1]$
and
$S_1=[0,1/3] \cup [2/3,1]$
and so on. You can see that the points in
$S_1$
comprise
$2/3$
of the points in
$S_0$: more precisely, $[0,1/3]$ has $1/3$ of the points in $S_0$ and $[2/3,1]$ has $1/3$ of the points in $S_0$. Here let $D$ be the average distance between any two points in the Cantor set $C$. Note that we are assuming the distance between two points in the last step of the construction, namely when we have no intervals any more, where we have a rational set. Now, if we pick any two points randomly, the chance that the two points are in the same half is $1/2$ (clearly also $1/2$ that they are in different halves). If the two points are in the same half, we get the average distance $D/3$ (which means third of the length of the step (interval) before) and if the two points in different halves we have the average distance as $2/3$.
So the average distance satisfies the equation
$$D = \frac 1 2 \cdot\frac D 3 + \frac 1 2 \cdot\frac 2 3,$$
namely $D=D/6+1/3$ and so $D=2/5.$
