Some Generalizations of Random Broken and Pick-up Stick Problems

Abstract

Abstract We consider sticks with random lengths, which are further randomly broken into several pieces. The probability that these sticks can form a polygon is computed in this article. This Broken Pick-up Stick Problem was first asked in [1] and we give a general formula to solve this problem. Besides, we study the probability that any three sticks with independent and uniformly distributed lengths can form a triangle. Our results generalize the famous Spaghetti Problem and the Pick-up Stick Problem in probability. We present a way to transfer these continuous probability problems into discrete problems and apply combinatorial methods to address the discrete version of the problems.