This post is inspired by a comment from Nuzzle in “Glass Rod Problem”, http://forums.codeguru.com/showthrea...47#post2073747. Internet searches show that the problem is more often called the “Broken Stick Problem”, which is what I’ve called it here.

The basic problem: If a stick is broken randomly into three pieces, what is the probability that the three pieces can be arranged end-to-end to form a triangle?

My issue is that I get different answers depending on how I phrase the problem:

Problem Statement #1: break a stick into three pieces by randomly selecting two break points. What is the probability that the three pieces can be arranged end-to-end to form a triangle?

Answer to #1: 0.25, as verified by countless answers on the web, and by the attached Excel spreadsheet showing a Monte Carlo simulation. The spreadsheet uses the same basic logic as that given by Nuzzle in his above post.

Problem Statement #2: break a stick into two pieces and then randomly select one of the pieces. Break the selected piece again, to form a total of three pieces. What is the probability that the three pieces can be arranged end-to-end to form a triangle?

Answer to #2: 0.193147 which is equal to ln(2) - 0.5 (which is the integral of (1-x)/x from .5 to 1 -- don’t ask).

Huh? Problems #1 and #2 seem identical to me, but they give dramatically different answers. Why is that?

And to confuse matters even more ...

Problem Statement #3: break a stick into two pieces and then select one of the pieces, but not randomly. Instead, select a piece with probability proportional to length, so that the longer piece is selected more often than the shorter one. Break the selected piece again, to form a total of three pieces. What is the probability that the three pieces can be arranged end-to-end to form a triangle?

Answer to #3: 0.25.

Once again, the result is puzzling. Problems #1 and #3 seem completely different, yet they both yield the exact same answer. Why is that?

The attached Excel spreadsheet shows Monte Carlo simulations of all three statements of the problem. To determine whether a triangle can be formed, we use the triangle inequality, which (in one form) basically says that in a triangle, the longest side is smaller than the sum of the two shorter sides. In the spreadsheet, the length of the stick is taken as “1”, so with a little algebra, we can determine that a triangle can be formed so long as MAX(all three sides) < 0.5.

Can anyone explain the reasons for the above results?

Mike

Broken-Stick-Problem.zip