my strictly amateur guess would go as follows;
replace the stick with line segment AD. Now select at random 1 point (B)on line segment AD. Now we have 3 possibilities.
1. AB > BD
2. AB < BD
3. AB = BD
In an infinite number of selections, avg(AB) = avg(BD).

Now select a 2nd random point (C) on line segment AD.
Clearly it must fall between A and B or between B and D.
So,
Since avg(AB) = avg(BD), in an infinite number of selections, point C should fall between A and B 50% of the time, and between B and D 50% of the time.

Now, for every case, we have AD = AB+BC+CD or AD = AC+CB+BD.
Also we can only make a triangle if no segment >1/2 AD.

In virtually every case, AB<>BD, so either AB or BD >1/2 AD. So, if point C falls equally on either side of point B, it must, as the number of trials approaches infinity, fall half the time on the shorter segment, leaving the other segment >1/2 AD. Therefore I would conclude the answer to be 1/2, or .5.

I would appreciate if someone could confirm this result, or demonstrate an error in my logic.
Thank You.