Random Chords

One of the most common class of probabilistic questions concerns
"random triangles".  For example, there's a little Dover book 
called "The Surprise Attack in Mathematical Problems" by L. A. 
Graham, consisting of a collection of problems and solutions 
that appeared in Graham's newspaper column over the years.  One 
whole chapter is devoted to the problem "What is the chance that 
the altitudes of a triangle may themselves form another triangle?"  
In reviewing the range of answers and arguments that were 
prompted by this question, Graham dryly notes that "the choice 
of the optimum answer introduces matters of philosophy, esthetics, 
symmetry, and even psychiatry".

A similar problem is to determine the probability that a randomly 
selected chord of a regular n-gon (n>3) is shorter than the side of 
the n-gon.  Of course this is a variation of a familiar class of 
problems, such as finding the probability that a "random chord" of 
a circle is longer than the radius, and as with all such problems it 
clearly depends on the assumed distribution.  One common definition
of a "random chord" is to assume that the endpoints of the chord 
are uniformly distributed on the perimeter of the polygon.  On this 
basis, there's a 1/n probability of the two ends of the chord falling 
on the same edge, and a 2/n probability of falling on adjacent edges, 
in which case the probability of the chord being shorter than an edge 
length is just the area in the first quadrant inside the ellipse

                 x^2 + 2xycos(2pi/n) + y^2 = 1

giving an overall probability of 

                   1   /       2pi/n    \
                  --- ( 1 + -----------  )
                   n   \     sin(2pi/n) /

Of course, other assumptions as to the distribution of "random chords"
will give different answers.