Disjoint Placement Probability of Line Segments via Induction

Abstract

We give a different proof of the following recent result: Let a finite number n of line segments, the sum L_n of whose lengths is less than one, be placed onto the real line in such a way that their centers fall randomly within the unit interval $[0,1]$. Then the probability of obtaining a mutually disjoint placement of these segments, entirely within $[0,1]$, is given by ${(1-L_n)}^n$. The proof presented here uses induction on the number of line segments and provides insight, at each level of the induction, into the relationship between two seemingly different methods of placement: sequential random placement versus simultaneous random placement. From a purely mathematical perspective, these methods can be seen as equivalent. However, physical constraints in performing a simultaneous random placement of actual segments (e.g., toothpicks) might a priori lead to very different outcomes.

Acknowledgment

The authors express their sincere appreciation to the referees and the Editor for their many helpful suggestions. These have greatly improved the clarity and readability of the article. The proof concept here is due to IH and the exposition is due to CE.

Additional information

Notes on contributors

Christopher Ennis

Christopher Ennis ([email protected]) attended UCLA and UC Berkeley. After holding several academic appointments, he settled into a rewarding 24 year teaching career at Normandale Community College, including several years as department chairperson. Retiring in 2016, he has again found time to do mathematical research.

Inge Helland

Inge Helland ([email protected]) is a retired professor of statistics from the University of Oslo. After retirement he has spent most of his time trying to understand the foundation of quantum mechanics from his point of view as a statistician. This has resulted in two books and several articles. But he also has some interest in recreational mathematics.