
Re: Random Triangle Problem (LONG summary)
Posted:
Aug 16, 1997 3:22 PM


On Fri, 08 Aug 1997 T.Moore@massey.ac.nz (Terry Moore) wrote: >Keith Ramsay has drawn our attention to an American >Mathematical Monthly article which has discussed this >question and looked at several distributions for the >vertices.
In addition to that article, the same and similar questions about "random triangles" have been discussed in print and in this newsgroup several times before. 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".
Coincidentally, I recently got an email from two college students who had determined THE probability that a randomly selected chord of a regular ngon (n>3) is shorter than the side of the ngon, and they were wondering if perhaps they were due a Fields Medal. 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. The two guys were quite disappointed to learn that the definition of a "random chord" they were using (which they had taken from a Cliff Pickover book) was not universally regarded as THE unique definition of a random chord, and that other equally plausible definitions give different answers.
Anyway, the assumption they were using was that the endpoints of the chord are uniformly distributed on the perimeter of the polygon. They chose not to reveal their solution (thinking there might still be big money in it), but I imagine they reasoned that 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) /

